| L(s) = 1 | − 4·2-s + 2·3-s + 12·4-s − 8·6-s − 32·8-s − 5·9-s − 14·11-s + 24·12-s + 80·16-s − 4·17-s + 20·18-s + 56·22-s − 64·24-s − 50·25-s − 28·27-s − 192·32-s − 28·33-s + 16·34-s − 60·36-s − 92·41-s − 168·44-s + 160·48-s − 98·49-s + 200·50-s − 8·51-s + 112·54-s + 448·64-s + ⋯ |
| L(s) = 1 | − 2·2-s + 2/3·3-s + 3·4-s − 4/3·6-s − 4·8-s − 5/9·9-s − 1.27·11-s + 2·12-s + 5·16-s − 0.235·17-s + 10/9·18-s + 2.54·22-s − 8/3·24-s − 2·25-s − 1.03·27-s − 6·32-s − 0.848·33-s + 8/17·34-s − 5/3·36-s − 2.24·41-s − 3.81·44-s + 10/3·48-s − 2·49-s + 4·50-s − 0.156·51-s + 2.07·54-s + 7·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s+1)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.4313482404\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4313482404\) |
| \(L(2)\) |
|
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 | $C_1$ | \( ( 1 + p T )^{2} \) |
| 3 | $C_2$ | \( 1 - 2 T + p^{2} T^{2} \) |
| 11 | $C_2$ | \( 1 + 14 T + p^{2} T^{2} \) |
| good | 5 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 7 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 13 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 17 | $C_2$ | \( ( 1 + 2 T + p^{2} T^{2} )^{2} \) |
| 19 | $C_2$ | \( ( 1 - 34 T + p^{2} T^{2} )( 1 + 34 T + p^{2} T^{2} ) \) |
| 23 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 29 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 31 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 37 | $C_1$$\times$$C_1$ | \( ( 1 - p T )^{2}( 1 + p T )^{2} \) |
| 41 | $C_2$ | \( ( 1 + 46 T + p^{2} T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 - 14 T + p^{2} T^{2} )( 1 + 14 T + p^{2} T^{2} ) \) |
| 47 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 53 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 59 | $C_2$ | \( ( 1 - 82 T + p^{2} T^{2} )( 1 + 82 T + p^{2} T^{2} ) \) |
| 61 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 62 T + p^{2} T^{2} )^{2} \) |
| 71 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 142 T + p^{2} T^{2} )( 1 + 142 T + p^{2} T^{2} ) \) |
| 79 | $C_2$ | \( ( 1 + p^{2} T^{2} )^{2} \) |
| 83 | $C_2$ | \( ( 1 - 158 T + p^{2} T^{2} )^{2} \) |
| 89 | $C_2$ | \( ( 1 - 146 T + p^{2} T^{2} )( 1 + 146 T + p^{2} T^{2} ) \) |
| 97 | $C_2$ | \( ( 1 - 94 T + p^{2} T^{2} )^{2} \) |
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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
−11.80336749437518193559562541059, −11.32627065978106592864861146579, −10.91779064561977793638239571376, −10.32865806456957640401268573534, −9.987606708717199599693871560417, −9.552767115905468611488245600374, −9.132922022417845522915308446752, −8.391440902035262872083499578145, −8.321587562292204631537915704994, −7.68845976713507922485247517972, −7.54175650456623042283414451699, −6.62536571140026943070190178960, −6.25786266993057920231026535832, −5.57392090263471256081235437610, −4.98341732910305405642762075367, −3.47699813429912053767108011948, −3.26241689648122793973265438821, −2.09435769287856637016964126806, −2.05531474876567140283238055154, −0.41410663730054621594562737847,
0.41410663730054621594562737847, 2.05531474876567140283238055154, 2.09435769287856637016964126806, 3.26241689648122793973265438821, 3.47699813429912053767108011948, 4.98341732910305405642762075367, 5.57392090263471256081235437610, 6.25786266993057920231026535832, 6.62536571140026943070190178960, 7.54175650456623042283414451699, 7.68845976713507922485247517972, 8.321587562292204631537915704994, 8.391440902035262872083499578145, 9.132922022417845522915308446752, 9.552767115905468611488245600374, 9.987606708717199599693871560417, 10.32865806456957640401268573534, 10.91779064561977793638239571376, 11.32627065978106592864861146579, 11.80336749437518193559562541059
