| L(s) = 1 | + 2·2-s + 3-s + 3·4-s + 2·5-s + 2·6-s − 2·7-s + 4·8-s − 4·9-s + 4·10-s − 10·11-s + 3·12-s + 3·13-s − 4·14-s + 2·15-s + 5·16-s + 7·17-s − 8·18-s + 6·20-s − 2·21-s − 20·22-s − 12·23-s + 4·24-s − 7·25-s + 6·26-s − 6·27-s − 6·28-s − 12·29-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 0.577·3-s + 3/2·4-s + 0.894·5-s + 0.816·6-s − 0.755·7-s + 1.41·8-s − 4/3·9-s + 1.26·10-s − 3.01·11-s + 0.866·12-s + 0.832·13-s − 1.06·14-s + 0.516·15-s + 5/4·16-s + 1.69·17-s − 1.88·18-s + 1.34·20-s − 0.436·21-s − 4.26·22-s − 2.50·23-s + 0.816·24-s − 7/5·25-s + 1.17·26-s − 1.15·27-s − 1.13·28-s − 2.22·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 25542916 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25542916 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( ( 1 - T )^{2} \) |
| 7 | $C_1$ | \( ( 1 + T )^{2} \) |
| 19 | | \( 1 \) |
| good | 3 | $D_{4}$ | \( 1 - T + 5 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 5 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 11 | $C_2$ | \( ( 1 + 5 T + p T^{2} )^{2} \) |
| 13 | $D_{4}$ | \( 1 - 3 T + 17 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $D_{4}$ | \( 1 - 7 T + 35 T^{2} - 7 p T^{3} + p^{2} T^{4} \) |
| 23 | $D_{4}$ | \( 1 + 12 T + 77 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 29 | $D_{4}$ | \( 1 + 12 T + 89 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 31 | $C_2^2$ | \( 1 + 17 T^{2} + p^{2} T^{4} \) |
| 37 | $D_{4}$ | \( 1 + 6 T + 3 T^{2} + 6 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_4$ | \( 1 - 9 T + 71 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 43 | $D_{4}$ | \( 1 + 7 T + 87 T^{2} + 7 p T^{3} + p^{2} T^{4} \) |
| 47 | $D_{4}$ | \( 1 + 2 T + 15 T^{2} + 2 p T^{3} + p^{2} T^{4} \) |
| 53 | $D_{4}$ | \( 1 + 12 T + 97 T^{2} + 12 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 11 T + p T^{2} )^{2} \) |
| 61 | $D_{4}$ | \( 1 - 6 T + 111 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 67 | $D_{4}$ | \( 1 - 9 T + 143 T^{2} - 9 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 73 | $D_{4}$ | \( 1 + 9 T + 65 T^{2} + 9 p T^{3} + p^{2} T^{4} \) |
| 79 | $D_{4}$ | \( 1 + 11 T + 177 T^{2} + 11 p T^{3} + p^{2} T^{4} \) |
| 83 | $D_{4}$ | \( 1 + 4 T - 10 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 89 | $D_{4}$ | \( 1 - 3 T + 169 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 97 | $D_{4}$ | \( 1 + 10 T + 199 T^{2} + 10 p T^{3} + p^{2} T^{4} \) |
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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
−7.894485621678093162929055563888, −7.88715610108539321239776415825, −7.28302754501355177556202419465, −7.06845957627723859894693942724, −6.24310103125100703018274333698, −5.96780323568502429672394527833, −5.67776377120205288220024769371, −5.66270192552395995127594064525, −5.34975056671965840447134039218, −4.98961717681091109412073528918, −4.02620808779089183486358775718, −3.98159073182168572648894385208, −3.34746484326330225806900423569, −3.24512899773144223555405793387, −2.58307551410786850409947326964, −2.44152619398637446730108803534, −1.98765026880988300524314540212, −1.52007451308014621548898381245, 0, 0,
1.52007451308014621548898381245, 1.98765026880988300524314540212, 2.44152619398637446730108803534, 2.58307551410786850409947326964, 3.24512899773144223555405793387, 3.34746484326330225806900423569, 3.98159073182168572648894385208, 4.02620808779089183486358775718, 4.98961717681091109412073528918, 5.34975056671965840447134039218, 5.66270192552395995127594064525, 5.67776377120205288220024769371, 5.96780323568502429672394527833, 6.24310103125100703018274333698, 7.06845957627723859894693942724, 7.28302754501355177556202419465, 7.88715610108539321239776415825, 7.894485621678093162929055563888
