| L(s) = 1 | + 2-s + 3-s + 4-s + 2.82·5-s + 6-s + 8-s + 9-s + 2.82·10-s + 2·11-s + 12-s − 2.82·13-s + 2.82·15-s + 16-s + 18-s − 19-s + 2.82·20-s + 2·22-s − 0.828·23-s + 24-s + 3.00·25-s − 2.82·26-s + 27-s + 7.65·29-s + 2.82·30-s − 6.82·31-s + 32-s + 2·33-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s + 0.5·4-s + 1.26·5-s + 0.408·6-s + 0.353·8-s + 0.333·9-s + 0.894·10-s + 0.603·11-s + 0.288·12-s − 0.784·13-s + 0.730·15-s + 0.250·16-s + 0.235·18-s − 0.229·19-s + 0.632·20-s + 0.426·22-s − 0.172·23-s + 0.204·24-s + 0.600·25-s − 0.554·26-s + 0.192·27-s + 1.42·29-s + 0.516·30-s − 1.22·31-s + 0.176·32-s + 0.348·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5586 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.360906698\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.360906698\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 5 | \( 1 - 2.82T + 5T^{2} \) |
| 11 | \( 1 - 2T + 11T^{2} \) |
| 13 | \( 1 + 2.82T + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 23 | \( 1 + 0.828T + 23T^{2} \) |
| 29 | \( 1 - 7.65T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 + 3.65T + 41T^{2} \) |
| 43 | \( 1 - 8T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 + 3.65T + 53T^{2} \) |
| 59 | \( 1 - 9.65T + 59T^{2} \) |
| 61 | \( 1 - 13.3T + 61T^{2} \) |
| 67 | \( 1 - 10T + 67T^{2} \) |
| 71 | \( 1 + 9.65T + 71T^{2} \) |
| 73 | \( 1 - 0.343T + 73T^{2} \) |
| 79 | \( 1 + 3.17T + 79T^{2} \) |
| 83 | \( 1 + 11.3T + 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−8.115640302793372083477420152088, −7.26144903926484608163158747612, −6.62718645176683169687194761676, −5.95377953125461432720214886114, −5.27979585388743941557263992936, −4.46211429403156860998865003224, −3.72389786789076681912868110037, −2.62372392554141827030980023453, −2.21776092294904189227348202467, −1.16217364667712924837353049275,
1.16217364667712924837353049275, 2.21776092294904189227348202467, 2.62372392554141827030980023453, 3.72389786789076681912868110037, 4.46211429403156860998865003224, 5.27979585388743941557263992936, 5.95377953125461432720214886114, 6.62718645176683169687194761676, 7.26144903926484608163158747612, 8.115640302793372083477420152088
