| L(s) = 1 | + 2-s + 3·3-s + 2·4-s + 3·5-s + 3·6-s − 7-s + 5·8-s + 3·9-s + 3·10-s − 3·11-s + 6·12-s − 14-s + 9·15-s + 5·16-s + 2·17-s + 3·18-s − 19-s + 6·20-s − 3·21-s − 3·22-s + 15·24-s + 5·25-s − 2·28-s + 14·29-s + 9·30-s + 3·31-s + 10·32-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.73·3-s + 4-s + 1.34·5-s + 1.22·6-s − 0.377·7-s + 1.76·8-s + 9-s + 0.948·10-s − 0.904·11-s + 1.73·12-s − 0.267·14-s + 2.32·15-s + 5/4·16-s + 0.485·17-s + 0.707·18-s − 0.229·19-s + 1.34·20-s − 0.654·21-s − 0.639·22-s + 3.06·24-s + 25-s − 0.377·28-s + 2.59·29-s + 1.64·30-s + 0.538·31-s + 1.76·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399489 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399489 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(10.26166505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(10.26166505\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 7 | $C_2$ | \( 1 + T + p T^{2} \) |
| 13 | | \( 1 \) |
| good | 2 | $C_2^2$ | \( 1 - T - T^{2} - p T^{3} + p^{2} T^{4} \) |
| 3 | $C_2$ | \( ( 1 - p T + p T^{2} )( 1 + p T^{2} ) \) |
| 5 | $C_2^2$ | \( 1 - 3 T + 4 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 11 | $C_2^2$ | \( 1 + 3 T - 2 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 - 2 T - 13 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 - 7 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 23 | $C_2^2$ | \( 1 - p T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 7 T + p T^{2} )^{2} \) |
| 31 | $C_2^2$ | \( 1 - 3 T - 22 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 37 | $C_2^2$ | \( 1 - 2 T - 33 T^{2} - 2 p T^{3} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 3 T + p T^{2} )^{2} \) |
| 43 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - T - 46 T^{2} - p T^{3} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 + 3 T - 44 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 59 | $C_2^2$ | \( 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 14 T + p T^{2} )( 1 + T + p T^{2} ) \) |
| 67 | $C_2^2$ | \( 1 + 3 T - 58 T^{2} + 3 p T^{3} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 13 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 13 T + 96 T^{2} + 13 p T^{3} + p^{2} T^{4} \) |
| 79 | $C_2^2$ | \( 1 - 3 T - 70 T^{2} - 3 p T^{3} + p^{2} T^{4} \) |
| 83 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 89 | $C_2^2$ | \( 1 - 6 T - 53 T^{2} - 6 p T^{3} + p^{2} T^{4} \) |
| 97 | $C_2$ | \( ( 1 - 5 T + p T^{2} )^{2} \) |
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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
−9.967938081850056634958906520255, −9.915139081518633209634128081390, −9.012282769476694292121797395843, −8.548154181270217259632686312404, −8.474630445627107355942521446412, −7.983749195653116349503547621320, −7.50956611988400964464052029267, −7.10404181606467207098691996136, −6.61751085047204545254630322960, −6.21660831718768723493081109880, −5.88025568174882864734349845140, −5.14764001109159023306996383105, −4.70989735262425109046900999124, −4.54059114014824286517521903492, −3.53701820624522614986763061884, −3.04104789468140083456004943034, −2.90056379113160664348048663499, −2.29101666298376808114116792943, −1.86504054729845776667317732501, −1.21463702303332294204962973448,
1.21463702303332294204962973448, 1.86504054729845776667317732501, 2.29101666298376808114116792943, 2.90056379113160664348048663499, 3.04104789468140083456004943034, 3.53701820624522614986763061884, 4.54059114014824286517521903492, 4.70989735262425109046900999124, 5.14764001109159023306996383105, 5.88025568174882864734349845140, 6.21660831718768723493081109880, 6.61751085047204545254630322960, 7.10404181606467207098691996136, 7.50956611988400964464052029267, 7.983749195653116349503547621320, 8.474630445627107355942521446412, 8.548154181270217259632686312404, 9.012282769476694292121797395843, 9.915139081518633209634128081390, 9.967938081850056634958906520255
