L(s) = 1 | + (−0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)6-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (−0.809 − 0.587i)11-s + (−0.309 + 0.951i)12-s + (0.309 − 0.951i)16-s + (−0.5 + 1.53i)17-s − 18-s + (−0.690 + 0.951i)19-s + (−0.309 + 0.951i)22-s + 24-s + (0.809 + 0.587i)25-s + ⋯ |
L(s) = 1 | + (−0.309 − 0.951i)2-s + (0.809 − 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.809 − 0.587i)6-s + (0.809 + 0.587i)8-s + (0.309 − 0.951i)9-s + (−0.809 − 0.587i)11-s + (−0.309 + 0.951i)12-s + (0.309 − 0.951i)16-s + (−0.5 + 1.53i)17-s − 18-s + (−0.690 + 0.951i)19-s + (−0.309 + 0.951i)22-s + 24-s + (0.809 + 0.587i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 264 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0219 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7448744236\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7448744236\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.309 + 0.951i)T \) |
| 3 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 + (0.809 + 0.587i)T \) |
good | 5 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 7 | \( 1 + (0.309 - 0.951i)T^{2} \) |
| 13 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 17 | \( 1 + (0.5 - 1.53i)T + (-0.809 - 0.587i)T^{2} \) |
| 19 | \( 1 + (0.690 - 0.951i)T + (-0.309 - 0.951i)T^{2} \) |
| 23 | \( 1 + T^{2} \) |
| 29 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 31 | \( 1 + (0.809 - 0.587i)T^{2} \) |
| 37 | \( 1 + (-0.309 + 0.951i)T^{2} \) |
| 41 | \( 1 + (-0.5 - 0.363i)T + (0.309 + 0.951i)T^{2} \) |
| 43 | \( 1 + 1.90iT - T^{2} \) |
| 47 | \( 1 + (0.309 + 0.951i)T^{2} \) |
| 53 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 59 | \( 1 + (-0.690 - 0.951i)T + (-0.309 + 0.951i)T^{2} \) |
| 61 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 67 | \( 1 + 1.61T + T^{2} \) |
| 71 | \( 1 + (-0.809 - 0.587i)T^{2} \) |
| 73 | \( 1 + (1.11 + 1.53i)T + (-0.309 + 0.951i)T^{2} \) |
| 79 | \( 1 + (-0.809 + 0.587i)T^{2} \) |
| 83 | \( 1 + (0.190 - 0.587i)T + (-0.809 - 0.587i)T^{2} \) |
| 89 | \( 1 + 1.90iT - T^{2} \) |
| 97 | \( 1 + (0.190 + 0.587i)T + (-0.809 + 0.587i)T^{2} \) |
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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
−12.13133516941438571246730536272, −10.86091215368286987168982275315, −10.21374712096993907237720058249, −8.898843340642144988924113244754, −8.381262672885333741108860267326, −7.42690522768475181477935584248, −5.92767411638311330883655343041, −4.18448420906802506211289394256, −3.07551707817062936015930334639, −1.77748988568706869591553957104,
2.63113915452359507356127308308, 4.44537801186332158620406808990, 5.11224072666187724035063062674, 6.74568507957364686157660971202, 7.61412346984387877567512652406, 8.588546237557443705435081948000, 9.379005822472893451831460728588, 10.18318432359784003586591843591, 11.14730586106035320721555556532, 12.87038017099685329000905192664