| L(s) = 1 | − 4-s + 6·9-s − 2·11-s + 16-s − 8·19-s + 4·29-s + 16·31-s − 6·36-s − 12·41-s + 2·44-s − 49-s + 24·59-s + 12·61-s − 64-s − 16·71-s + 8·76-s + 27·81-s + 12·89-s − 12·99-s + 12·101-s − 12·109-s − 4·116-s + 3·121-s − 16·124-s + 127-s + 131-s + 137-s + ⋯ |
| L(s) = 1 | − 1/2·4-s + 2·9-s − 0.603·11-s + 1/4·16-s − 1.83·19-s + 0.742·29-s + 2.87·31-s − 36-s − 1.87·41-s + 0.301·44-s − 1/7·49-s + 3.12·59-s + 1.53·61-s − 1/8·64-s − 1.89·71-s + 0.917·76-s + 3·81-s + 1.27·89-s − 1.20·99-s + 1.19·101-s − 1.14·109-s − 0.371·116-s + 3/11·121-s − 1.43·124-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.611192692\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.611192692\) |
| \(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + T^{2} \) |
| 5 | | \( 1 \) |
| 7 | $C_2$ | \( 1 + T^{2} \) |
| 11 | $C_1$ | \( ( 1 + T )^{2} \) |
| good | 3 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 4 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 26 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 58 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 + 50 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 150 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 97 | $C_2^2$ | \( 1 + 2 T^{2} + p^{2} T^{4} \) |
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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
−8.479816326129258661163567921376, −8.355617627863584770411614809056, −8.077098416438319564800579860657, −7.65364476217957883871251322260, −6.93799753149081084040927248998, −6.83340271396909096252237603195, −6.75287921987580755101291724082, −6.04252496644771846010131421495, −5.81347292187717565263842944876, −5.02785724494196585416238050921, −4.85251568477068158071045941744, −4.50302819768858446607104191440, −4.21598695817485834159578869980, −3.72302765625376312485476285707, −3.35991804425803285052485583672, −2.54378592467189329101646827651, −2.32174603135251232514600516372, −1.69638798061122833127718617643, −1.07923888739619167603270866168, −0.54409941995117396302367316009,
0.54409941995117396302367316009, 1.07923888739619167603270866168, 1.69638798061122833127718617643, 2.32174603135251232514600516372, 2.54378592467189329101646827651, 3.35991804425803285052485583672, 3.72302765625376312485476285707, 4.21598695817485834159578869980, 4.50302819768858446607104191440, 4.85251568477068158071045941744, 5.02785724494196585416238050921, 5.81347292187717565263842944876, 6.04252496644771846010131421495, 6.75287921987580755101291724082, 6.83340271396909096252237603195, 6.93799753149081084040927248998, 7.65364476217957883871251322260, 8.077098416438319564800579860657, 8.355617627863584770411614809056, 8.479816326129258661163567921376
