L(s) = 1 | + (0.909 − 0.415i)2-s + (0.654 − 0.755i)4-s + (0.989 + 0.142i)7-s + (0.281 − 0.959i)8-s + (−0.540 + 0.841i)9-s + (0.898 + 0.334i)11-s + (0.959 − 0.281i)14-s + (−0.142 − 0.989i)16-s + (−0.142 + 0.989i)18-s + (0.956 − 0.0683i)22-s + (−0.841 + 0.540i)23-s + (−0.909 − 0.415i)25-s + (0.755 − 0.654i)28-s + (1.86 − 0.133i)29-s + (−0.540 − 0.841i)32-s + ⋯ |
L(s) = 1 | + (0.909 − 0.415i)2-s + (0.654 − 0.755i)4-s + (0.989 + 0.142i)7-s + (0.281 − 0.959i)8-s + (−0.540 + 0.841i)9-s + (0.898 + 0.334i)11-s + (0.959 − 0.281i)14-s + (−0.142 − 0.989i)16-s + (−0.142 + 0.989i)18-s + (0.956 − 0.0683i)22-s + (−0.841 + 0.540i)23-s + (−0.909 − 0.415i)25-s + (0.755 − 0.654i)28-s + (1.86 − 0.133i)29-s + (−0.540 − 0.841i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.835 + 0.549i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.331072135\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.331072135\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.909 + 0.415i)T \) |
| 7 | \( 1 + (-0.989 - 0.142i)T \) |
| 23 | \( 1 + (0.841 - 0.540i)T \) |
good | 3 | \( 1 + (0.540 - 0.841i)T^{2} \) |
| 5 | \( 1 + (0.909 + 0.415i)T^{2} \) |
| 11 | \( 1 + (-0.898 - 0.334i)T + (0.755 + 0.654i)T^{2} \) |
| 13 | \( 1 + (0.281 + 0.959i)T^{2} \) |
| 17 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 19 | \( 1 + (0.989 + 0.142i)T^{2} \) |
| 29 | \( 1 + (-1.86 + 0.133i)T + (0.989 - 0.142i)T^{2} \) |
| 31 | \( 1 + (-0.841 + 0.540i)T^{2} \) |
| 37 | \( 1 + (1.17 - 0.254i)T + (0.909 - 0.415i)T^{2} \) |
| 41 | \( 1 + (0.415 - 0.909i)T^{2} \) |
| 43 | \( 1 + (0.373 - 0.203i)T + (0.540 - 0.841i)T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + (0.418 + 0.559i)T + (-0.281 + 0.959i)T^{2} \) |
| 59 | \( 1 + (-0.281 - 0.959i)T^{2} \) |
| 61 | \( 1 + (-0.540 - 0.841i)T^{2} \) |
| 67 | \( 1 + (-0.133 + 0.0498i)T + (0.755 - 0.654i)T^{2} \) |
| 71 | \( 1 + (1.74 + 0.797i)T + (0.654 + 0.755i)T^{2} \) |
| 73 | \( 1 + (-0.142 + 0.989i)T^{2} \) |
| 79 | \( 1 + (0.258 + 1.80i)T + (-0.959 + 0.281i)T^{2} \) |
| 83 | \( 1 + (0.909 - 0.415i)T^{2} \) |
| 89 | \( 1 + (0.841 + 0.540i)T^{2} \) |
| 97 | \( 1 + (-0.415 + 0.909i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.983407723036528178706205923108, −8.168956491106167364659304858865, −7.45971871194161911297909143587, −6.46661806663138913820883551781, −5.77163126976237374100422402071, −4.88952852949723212795535388199, −4.39184249894459039662945121607, −3.35277689380182084305259184912, −2.24407418164890665112996362510, −1.52957552484683635548837714516,
1.48707813190165640912902884507, 2.71373748296059653506859003957, 3.72451357461931201072457034608, 4.33238969116068987693408342786, 5.27415888183065066900704951991, 6.05134199515572285398538835080, 6.66686939074427816812127545849, 7.48913218081985709796387351084, 8.457552049692128961097847846674, 8.704880466444809134130423523449