| L(s) = 1 | + 1.85·2-s + 1.43·4-s − 5-s − 4.90·7-s − 1.05·8-s − 1.85·10-s + 4.61·13-s − 9.08·14-s − 4.81·16-s + 3.76·17-s − 4.84·19-s − 1.43·20-s + 0.860·23-s + 25-s + 8.54·26-s − 7.01·28-s − 10.3·29-s + 7.81·31-s − 6.80·32-s + 6.97·34-s + 4.90·35-s − 4.67·37-s − 8.97·38-s + 1.05·40-s + 2.97·41-s + 0.907·43-s + 1.59·46-s + ⋯ |
| L(s) = 1 | + 1.30·2-s + 0.715·4-s − 0.447·5-s − 1.85·7-s − 0.373·8-s − 0.585·10-s + 1.27·13-s − 2.42·14-s − 1.20·16-s + 0.913·17-s − 1.11·19-s − 0.319·20-s + 0.179·23-s + 0.200·25-s + 1.67·26-s − 1.32·28-s − 1.92·29-s + 1.40·31-s − 1.20·32-s + 1.19·34-s + 0.829·35-s − 0.768·37-s − 1.45·38-s + 0.166·40-s + 0.464·41-s + 0.138·43-s + 0.234·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5445 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.282453012\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.282453012\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 1.85T + 2T^{2} \) |
| 7 | \( 1 + 4.90T + 7T^{2} \) |
| 13 | \( 1 - 4.61T + 13T^{2} \) |
| 17 | \( 1 - 3.76T + 17T^{2} \) |
| 19 | \( 1 + 4.84T + 19T^{2} \) |
| 23 | \( 1 - 0.860T + 23T^{2} \) |
| 29 | \( 1 + 10.3T + 29T^{2} \) |
| 31 | \( 1 - 7.81T + 31T^{2} \) |
| 37 | \( 1 + 4.67T + 37T^{2} \) |
| 41 | \( 1 - 2.97T + 41T^{2} \) |
| 43 | \( 1 - 0.907T + 43T^{2} \) |
| 47 | \( 1 - 13.2T + 47T^{2} \) |
| 53 | \( 1 + 5.13T + 53T^{2} \) |
| 59 | \( 1 - 12.5T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 - 4.26T + 67T^{2} \) |
| 71 | \( 1 - 5.13T + 71T^{2} \) |
| 73 | \( 1 - 4.61T + 73T^{2} \) |
| 79 | \( 1 - 0.843T + 79T^{2} \) |
| 83 | \( 1 + 5.75T + 83T^{2} \) |
| 89 | \( 1 + 1.40T + 89T^{2} \) |
| 97 | \( 1 - 8.54T + 97T^{2} \) |
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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
−8.153798074059349552674348019840, −7.03726081071566556136668192384, −6.58195983422190529695996290880, −5.86456630492520761726708724671, −5.45059878664150933828591519277, −4.15013021222600401388521250376, −3.78723290940588905866495478225, −3.21680006222864058862335229537, −2.35166843644661460367780381020, −0.63781020251627984940894761677,
0.63781020251627984940894761677, 2.35166843644661460367780381020, 3.21680006222864058862335229537, 3.78723290940588905866495478225, 4.15013021222600401388521250376, 5.45059878664150933828591519277, 5.86456630492520761726708724671, 6.58195983422190529695996290880, 7.03726081071566556136668192384, 8.153798074059349552674348019840
