| L(s) = 1 | + 2·3-s + 3·9-s + 2·11-s + 4·17-s − 8·19-s − 6·25-s + 4·27-s + 4·33-s − 12·41-s − 8·43-s + 2·49-s + 8·51-s − 16·57-s − 24·59-s − 8·67-s − 12·73-s − 12·75-s + 5·81-s − 8·83-s + 20·89-s − 28·97-s + 6·99-s − 8·107-s + 4·113-s + 3·121-s − 24·123-s + 127-s + ⋯ |
| L(s) = 1 | + 1.15·3-s + 9-s + 0.603·11-s + 0.970·17-s − 1.83·19-s − 6/5·25-s + 0.769·27-s + 0.696·33-s − 1.87·41-s − 1.21·43-s + 2/7·49-s + 1.12·51-s − 2.11·57-s − 3.12·59-s − 0.977·67-s − 1.40·73-s − 1.38·75-s + 5/9·81-s − 0.878·83-s + 2.11·89-s − 2.84·97-s + 0.603·99-s − 0.773·107-s + 0.376·113-s + 3/11·121-s − 2.16·123-s + 0.0887·127-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1115136 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1115136 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | | \( 1 \) |
| 3 | $C_1$ | \( ( 1 - T )^{2} \) |
| 11 | $C_1$ | \( ( 1 - T )^{2} \) |
| good | 5 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 13 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 17 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 29 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 31 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 - 6 T + p T^{2} )( 1 + 6 T + p T^{2} ) \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 47 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 53 | $C_2$ | \( ( 1 - 2 T + p T^{2} )( 1 + 2 T + p T^{2} ) \) |
| 59 | $C_2$ | \( ( 1 + 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + 14 T + p T^{2} ) \) |
| 67 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 - 12 T + p T^{2} )( 1 + 12 T + p T^{2} ) \) |
| 73 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 4 T + p T^{2} )( 1 + 4 T + p T^{2} ) \) |
| 83 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 + 14 T + p T^{2} )^{2} \) |
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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
−8.107869137440758896860513890796, −7.47499472452228970996968762649, −7.06858359073225212180441434607, −6.52682547819372513880633729063, −6.22756869331455955650272528651, −5.68093590142524294402459265505, −5.08660020809670672027476290340, −4.31872022414437645633139653031, −4.30977538214279742560651831455, −3.47701098944345637236797015672, −3.22350914665665535918625727885, −2.56194464227123448811095009052, −1.69573254513794375477006333754, −1.58990079624890863110029511527, 0,
1.58990079624890863110029511527, 1.69573254513794375477006333754, 2.56194464227123448811095009052, 3.22350914665665535918625727885, 3.47701098944345637236797015672, 4.30977538214279742560651831455, 4.31872022414437645633139653031, 5.08660020809670672027476290340, 5.68093590142524294402459265505, 6.22756869331455955650272528651, 6.52682547819372513880633729063, 7.06858359073225212180441434607, 7.47499472452228970996968762649, 8.107869137440758896860513890796
