| L(s) = 1 | + 2·3-s + 4·5-s − 6·7-s + 2·9-s + 8·15-s + 12·17-s − 12·21-s + 10·25-s + 6·27-s − 24·35-s + 12·37-s + 8·41-s − 16·43-s + 8·45-s + 8·47-s + 18·49-s + 24·51-s − 12·63-s + 48·67-s + 20·75-s + 11·81-s + 4·83-s + 48·85-s − 40·89-s − 32·101-s − 48·105-s + 24·111-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 1.78·5-s − 2.26·7-s + 2/3·9-s + 2.06·15-s + 2.91·17-s − 2.61·21-s + 2·25-s + 1.15·27-s − 4.05·35-s + 1.97·37-s + 1.24·41-s − 2.43·43-s + 1.19·45-s + 1.16·47-s + 18/7·49-s + 3.36·51-s − 1.51·63-s + 5.86·67-s + 2.30·75-s + 11/9·81-s + 0.439·83-s + 5.20·85-s − 4.23·89-s − 3.18·101-s − 4.68·105-s + 2.27·111-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{16} \cdot 3^{4} \cdot 5^{4} \cdot 7^{4}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{4} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(9.357035496\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.357035496\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | $C_2^2$ | \( 1 - 2 T + 2 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 5 | $C_1$ | \( ( 1 - T )^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| good | 11 | $C_2^2$ | \( ( 1 - 2 T^{2} + p^{2} T^{4} )^{2} \) |
| 13 | $D_4\times C_2$ | \( 1 - 40 T^{2} + 718 T^{4} - 40 p^{2} T^{6} + p^{4} T^{8} \) |
| 17 | $D_{4}$ | \( ( 1 - 6 T + 38 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 19 | $C_4\times C_2$ | \( 1 - 4 T^{2} - 554 T^{4} - 4 p^{2} T^{6} + p^{4} T^{8} \) |
| 23 | $C_2^2$ | \( ( 1 - 30 T^{2} + p^{2} T^{4} )^{2} \) |
| 29 | $D_4\times C_2$ | \( 1 - 24 T^{2} + 1646 T^{4} - 24 p^{2} T^{6} + p^{4} T^{8} \) |
| 31 | $D_4\times C_2$ | \( 1 - 64 T^{2} + 2446 T^{4} - 64 p^{2} T^{6} + p^{4} T^{8} \) |
| 37 | $D_{4}$ | \( ( 1 - 6 T + 78 T^{2} - 6 p T^{3} + p^{2} T^{4} )^{2} \) |
| 41 | $D_{4}$ | \( ( 1 - 4 T + 66 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 43 | $D_{4}$ | \( ( 1 + 8 T + 22 T^{2} + 8 p T^{3} + p^{2} T^{4} )^{2} \) |
| 47 | $D_{4}$ | \( ( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} )^{2} \) |
| 53 | $D_4\times C_2$ | \( 1 - 140 T^{2} + 9238 T^{4} - 140 p^{2} T^{6} + p^{4} T^{8} \) |
| 59 | $C_2^2$ | \( ( 1 + 98 T^{2} + p^{2} T^{4} )^{2} \) |
| 61 | $D_4\times C_2$ | \( 1 - 184 T^{2} + 15406 T^{4} - 184 p^{2} T^{6} + p^{4} T^{8} \) |
| 67 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{4} \) |
| 71 | $D_4\times C_2$ | \( 1 - 224 T^{2} + 22126 T^{4} - 224 p^{2} T^{6} + p^{4} T^{8} \) |
| 73 | $D_4\times C_2$ | \( 1 - 120 T^{2} + 12638 T^{4} - 120 p^{2} T^{6} + p^{4} T^{8} \) |
| 79 | $C_2^2$ | \( ( 1 + 78 T^{2} + p^{2} T^{4} )^{2} \) |
| 83 | $D_{4}$ | \( ( 1 - 2 T - 78 T^{2} - 2 p T^{3} + p^{2} T^{4} )^{2} \) |
| 89 | $D_{4}$ | \( ( 1 + 20 T + 258 T^{2} + 20 p T^{3} + p^{2} T^{4} )^{2} \) |
| 97 | $D_4\times C_2$ | \( 1 - 360 T^{2} + 51038 T^{4} - 360 p^{2} T^{6} + p^{4} T^{8} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{8} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.53863146887763680972279542725, −6.48829552887408984569885148206, −6.36115738539302846340126641886, −6.00009064071984630116538823024, −5.84840825225454576756091842410, −5.44900749260016857232631251356, −5.42289194262620725115761263088, −5.32133150645479107727527421396, −5.25627354718048348144185141231, −4.60861308150574951628305546906, −4.33347843379968837636844226024, −4.16458103178201432208284735876, −4.01510154844621858577533554741, −3.40450299375901995683680164459, −3.38335482347094237442928786093, −3.23089082113894238511429658890, −3.15775166153584044231725753821, −2.59616421798711717130358044920, −2.51259026344282328234957671307, −2.36840843839352153272681311307, −2.04396550345161337876269465063, −1.41791591023400859453726654348, −1.27405640238640040672855378292, −0.880072527525161258418069746331, −0.54148375537052537338035615179,
0.54148375537052537338035615179, 0.880072527525161258418069746331, 1.27405640238640040672855378292, 1.41791591023400859453726654348, 2.04396550345161337876269465063, 2.36840843839352153272681311307, 2.51259026344282328234957671307, 2.59616421798711717130358044920, 3.15775166153584044231725753821, 3.23089082113894238511429658890, 3.38335482347094237442928786093, 3.40450299375901995683680164459, 4.01510154844621858577533554741, 4.16458103178201432208284735876, 4.33347843379968837636844226024, 4.60861308150574951628305546906, 5.25627354718048348144185141231, 5.32133150645479107727527421396, 5.42289194262620725115761263088, 5.44900749260016857232631251356, 5.84840825225454576756091842410, 6.00009064071984630116538823024, 6.36115738539302846340126641886, 6.48829552887408984569885148206, 6.53863146887763680972279542725
