| L(s) = 1 | + 2.03·2-s + 2.12·4-s + 3.40·5-s − 3.47·7-s + 0.255·8-s + 6.91·10-s − 11-s + 1.35·13-s − 7.04·14-s − 3.73·16-s + 4.44·17-s − 3.01·19-s + 7.24·20-s − 2.03·22-s + 2.96·23-s + 6.60·25-s + 2.75·26-s − 7.37·28-s + 7.28·29-s + 6.63·31-s − 8.09·32-s + 9.03·34-s − 11.8·35-s + 9.62·37-s − 6.12·38-s + 0.871·40-s − 1.06·41-s + ⋯ |
| L(s) = 1 | + 1.43·2-s + 1.06·4-s + 1.52·5-s − 1.31·7-s + 0.0904·8-s + 2.18·10-s − 0.301·11-s + 0.375·13-s − 1.88·14-s − 0.933·16-s + 1.07·17-s − 0.692·19-s + 1.61·20-s − 0.433·22-s + 0.618·23-s + 1.32·25-s + 0.539·26-s − 1.39·28-s + 1.35·29-s + 1.19·31-s − 1.43·32-s + 1.54·34-s − 1.99·35-s + 1.58·37-s − 0.993·38-s + 0.137·40-s − 0.166·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.224601743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.224601743\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 - 2.03T + 2T^{2} \) |
| 5 | \( 1 - 3.40T + 5T^{2} \) |
| 7 | \( 1 + 3.47T + 7T^{2} \) |
| 13 | \( 1 - 1.35T + 13T^{2} \) |
| 17 | \( 1 - 4.44T + 17T^{2} \) |
| 19 | \( 1 + 3.01T + 19T^{2} \) |
| 23 | \( 1 - 2.96T + 23T^{2} \) |
| 29 | \( 1 - 7.28T + 29T^{2} \) |
| 31 | \( 1 - 6.63T + 31T^{2} \) |
| 37 | \( 1 - 9.62T + 37T^{2} \) |
| 41 | \( 1 + 1.06T + 41T^{2} \) |
| 43 | \( 1 - 6.68T + 43T^{2} \) |
| 47 | \( 1 + 5.41T + 47T^{2} \) |
| 53 | \( 1 - 7.07T + 53T^{2} \) |
| 59 | \( 1 + 1.73T + 59T^{2} \) |
| 67 | \( 1 - 1.27T + 67T^{2} \) |
| 71 | \( 1 - 9.73T + 71T^{2} \) |
| 73 | \( 1 + 4.46T + 73T^{2} \) |
| 79 | \( 1 - 8.03T + 79T^{2} \) |
| 83 | \( 1 - 8.97T + 83T^{2} \) |
| 89 | \( 1 + 3.98T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−7.988594345577758640998378009385, −6.79603987881781765862306724209, −6.35760503439574008906757872393, −5.95756270387896935435347759970, −5.27731926356431702722075530250, −4.54792815834295289941964888883, −3.60303333534665699321092700363, −2.82207653996977094742372753538, −2.40493568435408716162359083766, −0.980636799663440840342977561313,
0.980636799663440840342977561313, 2.40493568435408716162359083766, 2.82207653996977094742372753538, 3.60303333534665699321092700363, 4.54792815834295289941964888883, 5.27731926356431702722075530250, 5.95756270387896935435347759970, 6.35760503439574008906757872393, 6.79603987881781765862306724209, 7.988594345577758640998378009385
