| L(s) = 1 | − 1.56·3-s + 0.438·7-s − 0.561·9-s + 11-s − 7.12·13-s − 4.68·17-s − 5.56·19-s − 0.684·21-s − 7.12·23-s + 5.56·27-s + 4.43·29-s + 5.56·31-s − 1.56·33-s − 11.5·37-s + 11.1·39-s + 4.24·41-s + 5.12·43-s + 13.3·47-s − 6.80·49-s + 7.31·51-s + 2.68·53-s + 8.68·57-s + 7.12·59-s − 8.43·61-s − 0.246·63-s + 11.1·69-s + 8.68·71-s + ⋯ |
| L(s) = 1 | − 0.901·3-s + 0.165·7-s − 0.187·9-s + 0.301·11-s − 1.97·13-s − 1.13·17-s − 1.27·19-s − 0.149·21-s − 1.48·23-s + 1.07·27-s + 0.824·29-s + 0.998·31-s − 0.271·33-s − 1.90·37-s + 1.78·39-s + 0.663·41-s + 0.781·43-s + 1.95·47-s − 0.972·49-s + 1.02·51-s + 0.368·53-s + 1.15·57-s + 0.927·59-s − 1.08·61-s − 0.0310·63-s + 1.33·69-s + 1.03·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5844249690\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5844249690\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - T \) |
| good | 3 | \( 1 + 1.56T + 3T^{2} \) |
| 7 | \( 1 - 0.438T + 7T^{2} \) |
| 13 | \( 1 + 7.12T + 13T^{2} \) |
| 17 | \( 1 + 4.68T + 17T^{2} \) |
| 19 | \( 1 + 5.56T + 19T^{2} \) |
| 23 | \( 1 + 7.12T + 23T^{2} \) |
| 29 | \( 1 - 4.43T + 29T^{2} \) |
| 31 | \( 1 - 5.56T + 31T^{2} \) |
| 37 | \( 1 + 11.5T + 37T^{2} \) |
| 41 | \( 1 - 4.24T + 41T^{2} \) |
| 43 | \( 1 - 5.12T + 43T^{2} \) |
| 47 | \( 1 - 13.3T + 47T^{2} \) |
| 53 | \( 1 - 2.68T + 53T^{2} \) |
| 59 | \( 1 - 7.12T + 59T^{2} \) |
| 61 | \( 1 + 8.43T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 8.68T + 71T^{2} \) |
| 73 | \( 1 - 7.12T + 73T^{2} \) |
| 79 | \( 1 + 13.3T + 79T^{2} \) |
| 83 | \( 1 + 6T + 83T^{2} \) |
| 89 | \( 1 + 2.68T + 89T^{2} \) |
| 97 | \( 1 - 13.1T + 97T^{2} \) |
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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
−8.427370412818046300524677604411, −7.52026240966927507300108007564, −6.75341301550725319453827304694, −6.22693881847726816094513036010, −5.39818737583240999217374122112, −4.64427694116915238533390526040, −4.13749483317241520562489363237, −2.67488523374915295410037272961, −2.05243449248889277573363199902, −0.42222316507330107313446073579,
0.42222316507330107313446073579, 2.05243449248889277573363199902, 2.67488523374915295410037272961, 4.13749483317241520562489363237, 4.64427694116915238533390526040, 5.39818737583240999217374122112, 6.22693881847726816094513036010, 6.75341301550725319453827304694, 7.52026240966927507300108007564, 8.427370412818046300524677604411
