| L(s) = 1 | − 2-s + 4-s − 3·5-s − 8-s − 9-s + 3·10-s − 7·13-s + 16-s + 18-s − 3·20-s + 4·25-s + 7·26-s − 9·29-s − 32-s − 36-s + 11·37-s + 3·40-s + 6·41-s + 3·45-s − 49-s − 4·50-s − 7·52-s − 12·53-s + 9·58-s + 17·61-s + 64-s + 21·65-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 1.34·5-s − 0.353·8-s − 1/3·9-s + 0.948·10-s − 1.94·13-s + 1/4·16-s + 0.235·18-s − 0.670·20-s + 4/5·25-s + 1.37·26-s − 1.67·29-s − 0.176·32-s − 1/6·36-s + 1.80·37-s + 0.474·40-s + 0.937·41-s + 0.447·45-s − 1/7·49-s − 0.565·50-s − 0.970·52-s − 1.64·53-s + 1.18·58-s + 2.17·61-s + 1/8·64-s + 2.60·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1216800 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | $C_1$ | \( 1 + T \) |
| 3 | $C_2$ | \( 1 + T^{2} \) |
| 5 | $C_2$ | \( 1 + 3 T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + 7 T + p T^{2} \) |
| good | 7 | $C_2^2$ | \( 1 + T^{2} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 3 T + p T^{2} )( 1 + 3 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - 7 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$$\times$$C_2$ | \( ( 1 + 3 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2^2$ | \( 1 - 55 T^{2} + p^{2} T^{4} \) |
| 37 | $C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - T + p T^{2} ) \) |
| 41 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + p T^{2} ) \) |
| 43 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 47 | $C_2^2$ | \( 1 + 31 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 44 T^{2} + p^{2} T^{4} \) |
| 61 | $C_2$$\times$$C_2$ | \( ( 1 - 10 T + p T^{2} )( 1 - 7 T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 - 26 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2^2$ | \( 1 + 35 T^{2} + p^{2} T^{4} \) |
| 73 | $C_2$$\times$$C_2$ | \( ( 1 - 16 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 - 5 T + p T^{2} )( 1 + 5 T + p T^{2} ) \) |
| 83 | $C_2^2$ | \( 1 + 148 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$$\times$$C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 15 T + p T^{2} ) \) |
| 97 | $C_2$$\times$$C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 16 T + p T^{2} ) \) |
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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
−7.82451020595095970133566968640, −7.44203474984014006099014151673, −7.16822134982401620207899531132, −6.71557647674797165585006170954, −6.09862799998149224218924874677, −5.59836643825249761309284794153, −5.12553668452881373284929837001, −4.55446523910371041766348822471, −4.15778203907672684885602189452, −3.59767214885527792660960441344, −2.98201133625833971950912951738, −2.47581659775700771183633529380, −1.90547611207714553514661334469, −0.74830644685183970570992956731, 0,
0.74830644685183970570992956731, 1.90547611207714553514661334469, 2.47581659775700771183633529380, 2.98201133625833971950912951738, 3.59767214885527792660960441344, 4.15778203907672684885602189452, 4.55446523910371041766348822471, 5.12553668452881373284929837001, 5.59836643825249761309284794153, 6.09862799998149224218924874677, 6.71557647674797165585006170954, 7.16822134982401620207899531132, 7.44203474984014006099014151673, 7.82451020595095970133566968640
