| L(s) = 1 | − 3-s − 7-s − 9-s − 2·11-s − 4·13-s − 3·17-s − 19-s + 21-s + 2·23-s + 7·29-s − 9·31-s + 2·33-s − 37-s + 4·39-s + 4·41-s + 4·43-s − 18·47-s − 9·49-s + 3·51-s − 5·53-s + 57-s + 6·59-s + 17·61-s + 63-s − 4·67-s − 2·69-s + 5·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s − 1/3·9-s − 0.603·11-s − 1.10·13-s − 0.727·17-s − 0.229·19-s + 0.218·21-s + 0.417·23-s + 1.29·29-s − 1.61·31-s + 0.348·33-s − 0.164·37-s + 0.640·39-s + 0.624·41-s + 0.609·43-s − 2.62·47-s − 9/7·49-s + 0.420·51-s − 0.686·53-s + 0.132·57-s + 0.781·59-s + 2.17·61-s + 0.125·63-s − 0.488·67-s − 0.240·69-s + 0.593·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19360000 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
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 | | \( 1 \) |
| 5 | | \( 1 \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $D_{4}$ | \( 1 + T + 2 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 7 | $D_{4}$ | \( 1 + T + 10 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 13 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 17 | $D_{4}$ | \( 1 + 3 T + 32 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 19 | $D_{4}$ | \( 1 + T + 34 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 - 2 T + 30 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 - 7 T + 32 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 31 | $D_{4}$ | \( 1 + 9 T + 78 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + T + 36 T^{2} + p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 43 | $D_{4}$ | \( 1 - 4 T + 22 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 18 T + 158 T^{2} + 18 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 5 T + 108 T^{2} + 5 p T^{3} + p^{2} T^{4} \) |
| 59 | $D_{4}$ | \( 1 - 6 T + 110 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 17 T + 156 T^{2} - 17 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 + 4 T + 70 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 - 5 T + 110 T^{2} - 5 p T^{3} + p^{2} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 79 | $D_{4}$ | \( 1 - 6 T + 14 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 - 2 T + 150 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 + 21 T + 284 T^{2} + 21 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 6 T + 186 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.211190131222800297184374818206, −7.83538240963322482354672910392, −7.39077999820485137948806978873, −6.98740915933196438995220719601, −6.73381882651130133374520054038, −6.45545514581530355375928387403, −5.89789693368969449223423947458, −5.64824529533083379964463030670, −5.04085896379131696417103129396, −5.03525195676812671544748595816, −4.53291418318305388802537594968, −4.10486097283440690788133143950, −3.38199229562414632168039425063, −3.30010465706580550696489788356, −2.47313148056951808123692565966, −2.40126659460195423059275307912, −1.72503987713961388220287935393, −0.997790571195992237680315839671, 0, 0,
0.997790571195992237680315839671, 1.72503987713961388220287935393, 2.40126659460195423059275307912, 2.47313148056951808123692565966, 3.30010465706580550696489788356, 3.38199229562414632168039425063, 4.10486097283440690788133143950, 4.53291418318305388802537594968, 5.03525195676812671544748595816, 5.04085896379131696417103129396, 5.64824529533083379964463030670, 5.89789693368969449223423947458, 6.45545514581530355375928387403, 6.73381882651130133374520054038, 6.98740915933196438995220719601, 7.39077999820485137948806978873, 7.83538240963322482354672910392, 8.211190131222800297184374818206
