| L(s) = 1 | + 102·3-s + 6.18e3·7-s + 3.84e3·9-s + 1.45e4·13-s − 1.60e5·19-s + 6.31e5·21-s − 2.77e5·27-s + 8.71e5·31-s − 2.31e6·37-s + 1.48e6·39-s − 1.98e6·43-s + 1.71e7·49-s − 1.63e7·57-s + 3.87e7·61-s + 2.37e7·63-s + 5.60e7·67-s + 5.04e7·73-s − 1.26e8·79-s − 5.34e7·81-s + 9.02e7·91-s + 8.89e7·93-s − 3.91e7·97-s − 2.26e8·103-s + 3.48e6·109-s − 2.36e8·111-s + 5.60e7·117-s + 4.27e8·121-s + ⋯ |
| L(s) = 1 | + 1.25·3-s + 2.57·7-s + 0.585·9-s + 0.510·13-s − 1.23·19-s + 3.24·21-s − 0.521·27-s + 0.944·31-s − 1.23·37-s + 0.643·39-s − 0.579·43-s + 2.98·49-s − 1.55·57-s + 2.79·61-s + 1.50·63-s + 2.78·67-s + 1.77·73-s − 3.25·79-s − 1.24·81-s + 1.31·91-s + 1.18·93-s − 0.441·97-s − 2.01·103-s + 0.0247·109-s − 1.55·111-s + 0.299·117-s + 1.99·121-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 90000 ^{s/2} \, \Gamma_{\C}(s+4)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(7.932474518\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.932474518\) |
| \(L(5)\) |
|
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 | $C_2$ | \( 1 - 34 p T + p^{8} T^{2} \) |
| 5 | | \( 1 \) |
| good | 7 | $C_2$ | \( ( 1 - 442 p T + p^{8} T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 38857702 p T^{2} + p^{16} T^{4} \) |
| 13 | $C_2$ | \( ( 1 - 7294 T + p^{8} T^{2} )^{2} \) |
| 17 | $C_2^2$ | \( 1 - 10482174722 T^{2} + p^{16} T^{4} \) |
| 19 | $C_2$ | \( ( 1 + 80326 T + p^{8} T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 147132606722 T^{2} + p^{16} T^{4} \) |
| 29 | $C_2^2$ | \( 1 - 253546712162 T^{2} + p^{16} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 435914 T + p^{8} T^{2} )^{2} \) |
| 37 | $C_2$ | \( ( 1 + 1159298 T + p^{8} T^{2} )^{2} \) |
| 41 | $C_2^2$ | \( 1 - 8591852110082 T^{2} + p^{16} T^{4} \) |
| 43 | $C_2$ | \( ( 1 + 990266 T + p^{8} T^{2} )^{2} \) |
| 47 | $C_2^2$ | \( 1 - 2717384513282 T^{2} + p^{16} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 23095221504482 T^{2} + p^{16} T^{4} \) |
| 59 | $C_2^2$ | \( 1 - 291106443928802 T^{2} + p^{16} T^{4} \) |
| 61 | $C_2$ | \( ( 1 - 19369154 T + p^{8} T^{2} )^{2} \) |
| 67 | $C_2$ | \( ( 1 - 28024294 T + p^{8} T^{2} )^{2} \) |
| 71 | $C_2^2$ | \( 1 - 158683081351682 T^{2} + p^{16} T^{4} \) |
| 73 | $C_2$ | \( ( 1 - 25230142 T + p^{8} T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 + 63401398 T + p^{8} T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 2244210903661922 T^{2} + p^{16} T^{4} \) |
| 89 | $C_2^2$ | \( 1 - 1743003813196802 T^{2} + p^{16} T^{4} \) |
| 97 | $C_2$ | \( ( 1 + 19550306 T + p^{8} T^{2} )^{2} \) |
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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
−10.69683498742712370864921429788, −9.895659779779809603035935354277, −9.783795200047655280436759341458, −8.780887719021094481471882938454, −8.470893036895053132574764816385, −8.321743080623460372387514417700, −8.108777117339872235447184067200, −7.31063116861039681926828778247, −6.96114038239248374655945911587, −6.22603603898239204500113198966, −5.48040636818793116213886454291, −5.00774474093062371041263756235, −4.61642804401811501476199253229, −3.81845534663616833505716637945, −3.71003917387226886301377130139, −2.47900677297435028671095237831, −2.38523376114015837698033017889, −1.60431741632281449840869862345, −1.36686106972729144370659133387, −0.50265149447840587598523707148,
0.50265149447840587598523707148, 1.36686106972729144370659133387, 1.60431741632281449840869862345, 2.38523376114015837698033017889, 2.47900677297435028671095237831, 3.71003917387226886301377130139, 3.81845534663616833505716637945, 4.61642804401811501476199253229, 5.00774474093062371041263756235, 5.48040636818793116213886454291, 6.22603603898239204500113198966, 6.96114038239248374655945911587, 7.31063116861039681926828778247, 8.108777117339872235447184067200, 8.321743080623460372387514417700, 8.470893036895053132574764816385, 8.780887719021094481471882938454, 9.783795200047655280436759341458, 9.895659779779809603035935354277, 10.69683498742712370864921429788
