| L(s) = 1 | − 2·5-s − 4·9-s − 4·11-s + 4·13-s + 4·17-s + 2·19-s − 8·23-s + 3·25-s + 4·29-s + 12·37-s + 4·41-s + 8·45-s − 6·49-s + 12·53-s + 8·55-s + 8·61-s − 8·65-s + 16·67-s − 8·71-s − 4·73-s + 24·79-s + 7·81-s − 16·83-s − 8·85-s − 12·89-s − 4·95-s + 4·97-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 4/3·9-s − 1.20·11-s + 1.10·13-s + 0.970·17-s + 0.458·19-s − 1.66·23-s + 3/5·25-s + 0.742·29-s + 1.97·37-s + 0.624·41-s + 1.19·45-s − 6/7·49-s + 1.64·53-s + 1.07·55-s + 1.02·61-s − 0.992·65-s + 1.95·67-s − 0.949·71-s − 0.468·73-s + 2.70·79-s + 7/9·81-s − 1.75·83-s − 0.867·85-s − 1.27·89-s − 0.410·95-s + 0.406·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310400 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2310400 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.272614339\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.272614339\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 3 | $C_2^2$ | \( 1 + 4 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 11 | $D_{4}$ | \( 1 + 4 T + 18 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 13 | $D_{4}$ | \( 1 - 4 T + 28 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 4 T + 30 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 29 | $D_{4}$ | \( 1 - 4 T + 54 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $D_{4}$ | \( 1 - 12 T + 92 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 41 | $D_{4}$ | \( 1 - 4 T + 78 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
| 43 | $C_2^2$ | \( 1 + 78 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 22 T^{2} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 - 12 T + 92 T^{2} - 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 110 T^{2} + p^{2} T^{4} \) |
| 61 | $D_{4}$ | \( 1 - 8 T + 106 T^{2} - 8 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 16 T + 196 T^{2} - 16 p T^{3} + p^{2} T^{4} \) |
| 71 | $D_{4}$ | \( 1 + 8 T + 30 T^{2} + 8 p T^{3} + p^{2} T^{4} \) |
| 73 | $D_{4}$ | \( 1 + 4 T - 50 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 - 24 T + 294 T^{2} - 24 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 89 | $D_{4}$ | \( 1 + 12 T + 206 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 - 4 T + 180 T^{2} - 4 p T^{3} + p^{2} T^{4} \) |
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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
−9.678396174006840221426253358788, −9.279899583000994898532622480357, −8.546023845162878341187752307343, −8.497137364131764817345041504282, −8.002401702181419939965463193504, −7.901800735337600638582187594710, −7.41891593420163043447895604177, −6.90713023617083646657899514980, −6.14750277908761481683413296837, −6.11906296075235778613866779405, −5.44996691928555218222450854756, −5.35388322309231451874930470982, −4.55945215810712469391339379799, −4.18096256886026010519908576510, −3.49860584015460536087637147757, −3.36684813268620376838480250098, −2.51801468018039126786721732517, −2.39117902522840006617867281474, −1.16703527292918390729128663720, −0.50151682254725627847781573475,
0.50151682254725627847781573475, 1.16703527292918390729128663720, 2.39117902522840006617867281474, 2.51801468018039126786721732517, 3.36684813268620376838480250098, 3.49860584015460536087637147757, 4.18096256886026010519908576510, 4.55945215810712469391339379799, 5.35388322309231451874930470982, 5.44996691928555218222450854756, 6.11906296075235778613866779405, 6.14750277908761481683413296837, 6.90713023617083646657899514980, 7.41891593420163043447895604177, 7.901800735337600638582187594710, 8.002401702181419939965463193504, 8.497137364131764817345041504282, 8.546023845162878341187752307343, 9.279899583000994898532622480357, 9.678396174006840221426253358788
