L(s) = 1 | − 4-s − 2·5-s − 3·9-s + 16-s − 4·19-s + 2·20-s − 25-s + 2·29-s − 20·31-s + 3·36-s + 6·41-s + 6·45-s + 4·59-s + 18·61-s − 64-s + 12·71-s + 4·76-s + 20·79-s − 2·80-s − 14·89-s + 8·95-s + 100-s + 30·101-s − 10·109-s − 2·116-s − 22·121-s + 20·124-s + ⋯ |
L(s) = 1 | − 1/2·4-s − 0.894·5-s − 9-s + 1/4·16-s − 0.917·19-s + 0.447·20-s − 1/5·25-s + 0.371·29-s − 3.59·31-s + 1/2·36-s + 0.937·41-s + 0.894·45-s + 0.520·59-s + 2.30·61-s − 1/8·64-s + 1.42·71-s + 0.458·76-s + 2.25·79-s − 0.223·80-s − 1.48·89-s + 0.820·95-s + 1/10·100-s + 2.98·101-s − 0.957·109-s − 0.185·116-s − 2·121-s + 1.79·124-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 240100 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.6370952933\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6370952933\) |
\(L(\frac{3}{2})\) |
|
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 | $C_2$ | \( 1 + 2 T + p T^{2} \) |
| 7 | | \( 1 \) |
good | 3 | $C_2^2$ | \( 1 + p T^{2} + p^{2} T^{4} \) |
| 11 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 13 | $C_2^2$ | \( 1 - 22 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 - 8 T + p T^{2} )( 1 + 8 T + p T^{2} ) \) |
| 19 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 23 | $C_2^2$ | \( 1 - 45 T^{2} + p^{2} T^{4} \) |
| 29 | $C_2$ | \( ( 1 - T + p T^{2} )^{2} \) |
| 31 | $C_2$ | \( ( 1 + 10 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 - 3 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 - 61 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2^2$ | \( 1 - 30 T^{2} + p^{2} T^{4} \) |
| 53 | $C_2^2$ | \( 1 - 70 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 - 2 T + p T^{2} )^{2} \) |
| 61 | $C_2$ | \( ( 1 - 9 T + p T^{2} )^{2} \) |
| 67 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 - 6 T + p T^{2} )^{2} \) |
| 73 | $C_2^2$ | \( 1 - 46 T^{2} + p^{2} T^{4} \) |
| 79 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 - 85 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 + 7 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - p T^{2} )^{2} \) |
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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
−11.20625310417646513588492080102, −10.74763762730017023390043941011, −10.49506508414703429801356897994, −9.767877890927488897866430958994, −9.179472810046171780248909079025, −9.065063327136548304541031009421, −8.476055642707823217919944216769, −8.039708844602938549015913210554, −7.71749910702203284772542943788, −7.12006120424806721003646622153, −6.64714120899035296149152315967, −6.01428805669115130069272882460, −5.33256647568010760834851226251, −5.27117429103822672790357983891, −4.32131888863180932911347754335, −3.74464901646632227273354820514, −3.60656505690847192980830180824, −2.59266321899983891570887367014, −1.92285065984731881074930943091, −0.47493605841542549925847141317,
0.47493605841542549925847141317, 1.92285065984731881074930943091, 2.59266321899983891570887367014, 3.60656505690847192980830180824, 3.74464901646632227273354820514, 4.32131888863180932911347754335, 5.27117429103822672790357983891, 5.33256647568010760834851226251, 6.01428805669115130069272882460, 6.64714120899035296149152315967, 7.12006120424806721003646622153, 7.71749910702203284772542943788, 8.039708844602938549015913210554, 8.476055642707823217919944216769, 9.065063327136548304541031009421, 9.179472810046171780248909079025, 9.767877890927488897866430958994, 10.49506508414703429801356897994, 10.74763762730017023390043941011, 11.20625310417646513588492080102