| L(s) = 1 | + 2·2-s + 2·4-s − 6·5-s + 4·7-s − 18·9-s − 12·10-s + 12·11-s + 8·14-s − 4·16-s − 36·18-s + 52·19-s − 12·20-s + 24·22-s + 18·25-s + 8·28-s − 96·29-s − 28·31-s − 8·32-s − 24·35-s − 36·36-s + 74·37-s + 104·38-s − 18·41-s + 24·44-s + 108·45-s + 84·47-s + 8·49-s + ⋯ |
| L(s) = 1 | + 2-s + 1/2·4-s − 6/5·5-s + 4/7·7-s − 2·9-s − 6/5·10-s + 1.09·11-s + 4/7·14-s − 1/4·16-s − 2·18-s + 2.73·19-s − 3/5·20-s + 1.09·22-s + 0.719·25-s + 2/7·28-s − 3.31·29-s − 0.903·31-s − 1/4·32-s − 0.685·35-s − 36-s + 2·37-s + 2.73·38-s − 0.439·41-s + 6/11·44-s + 12/5·45-s + 1.78·47-s + 8/49·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 676 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.176724877\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.176724877\) |
| \(L(2)\) |
|
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 | $C_2$ | \( 1 - p T + p T^{2} \) |
| 13 | $C_2$ | \( 1 + p^{2} T^{2} \) |
| good | 3 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 5 | $C_2^2$ | \( 1 + 6 T + 18 T^{2} + 6 p^{2} T^{3} + p^{4} T^{4} \) |
| 7 | $C_2^2$ | \( 1 - 4 T + 8 T^{2} - 4 p^{2} T^{3} + p^{4} T^{4} \) |
| 11 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 542 T^{2} + p^{4} T^{4} \) |
| 19 | $C_2^2$ | \( 1 - 52 T + 1352 T^{2} - 52 p^{2} T^{3} + p^{4} T^{4} \) |
| 23 | $C_2^2$ | \( 1 - 482 T^{2} + p^{4} T^{4} \) |
| 29 | $C_2$ | \( ( 1 + 48 T + p^{2} T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 + 28 T + 392 T^{2} + 28 p^{2} T^{3} + p^{4} T^{4} \) |
| 37 | $C_1$$\times$$C_2$ | \( ( 1 - p T )^{2}( 1 + p^{2} T^{2} ) \) |
| 41 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 43 | $C_2^2$ | \( 1 - 2402 T^{2} + p^{4} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 84 T + 3528 T^{2} - 84 p^{2} T^{3} + p^{4} T^{4} \) |
| 53 | $C_2$ | \( ( 1 - 30 T + p^{2} T^{2} )^{2} \) |
| 59 | $C_2^2$ | \( 1 + 108 T + 5832 T^{2} + 108 p^{2} T^{3} + p^{4} T^{4} \) |
| 61 | $C_2$ | \( ( 1 + 18 T + p^{2} T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 + 44 T + 968 T^{2} + 44 p^{2} T^{3} + p^{4} T^{4} \) |
| 71 | $C_2^2$ | \( 1 - 12 T + 72 T^{2} - 12 p^{2} T^{3} + p^{4} T^{4} \) |
| 73 | $C_2^2$ | \( 1 - 34 T + 578 T^{2} - 34 p^{2} T^{3} + p^{4} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + 108 T + p^{2} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 156 T + 12168 T^{2} - 156 p^{2} T^{3} + p^{4} T^{4} \) |
| 89 | $C_2^2$ | \( 1 + 18 T + 162 T^{2} + 18 p^{2} T^{3} + p^{4} T^{4} \) |
| 97 | $C_2^2$ | \( 1 + 94 T + 4418 T^{2} + 94 p^{2} T^{3} + p^{4} T^{4} \) |
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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
−17.35449125857574115996580014573, −16.75075000834519460076311928876, −16.44712681731140219534814499481, −15.50405230739533236134390730981, −14.72775324191117174237516499556, −14.70294232960147805991572306730, −13.90300393988905715008427178251, −13.41921440535433854909104938686, −12.27189113398734597220008821566, −11.80008865402941539056754557416, −11.30287356242241452065891629106, −11.12671721156407251343402286710, −9.331167023136773648796773316813, −9.001743277187219891262692603270, −7.74904870654413418110336879363, −7.41588762584680208753980303552, −5.84602179849635622379922007877, −5.38569459593678042418592606655, −4.00527522059654166627719770843, −3.18651302706180249264320587061,
3.18651302706180249264320587061, 4.00527522059654166627719770843, 5.38569459593678042418592606655, 5.84602179849635622379922007877, 7.41588762584680208753980303552, 7.74904870654413418110336879363, 9.001743277187219891262692603270, 9.331167023136773648796773316813, 11.12671721156407251343402286710, 11.30287356242241452065891629106, 11.80008865402941539056754557416, 12.27189113398734597220008821566, 13.41921440535433854909104938686, 13.90300393988905715008427178251, 14.70294232960147805991572306730, 14.72775324191117174237516499556, 15.50405230739533236134390730981, 16.44712681731140219534814499481, 16.75075000834519460076311928876, 17.35449125857574115996580014573
