| L(s) = 1 | + 2.55·2-s + 4.53·4-s − 4.10·5-s − 4.64·7-s + 6.46·8-s − 10.5·10-s + 11-s + 0.666·13-s − 11.8·14-s + 7.46·16-s − 4.42·17-s + 6.64·19-s − 18.6·20-s + 2.55·22-s + 1.40·23-s + 11.8·25-s + 1.70·26-s − 21.0·28-s − 5.66·29-s + 5.31·31-s + 6.14·32-s − 11.3·34-s + 19.0·35-s + 4.06·37-s + 16.9·38-s − 26.5·40-s − 6.02·41-s + ⋯ |
| L(s) = 1 | + 1.80·2-s + 2.26·4-s − 1.83·5-s − 1.75·7-s + 2.28·8-s − 3.32·10-s + 0.301·11-s + 0.184·13-s − 3.17·14-s + 1.86·16-s − 1.07·17-s + 1.52·19-s − 4.16·20-s + 0.544·22-s + 0.293·23-s + 2.37·25-s + 0.334·26-s − 3.97·28-s − 1.05·29-s + 0.954·31-s + 1.08·32-s − 1.93·34-s + 3.22·35-s + 0.668·37-s + 2.75·38-s − 4.20·40-s − 0.941·41-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
| \(L(1)\) |
\(\approx\) |
\(3.316198782\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.316198782\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - 2.55T + 2T^{2} \) |
| 5 | \( 1 + 4.10T + 5T^{2} \) |
| 7 | \( 1 + 4.64T + 7T^{2} \) |
| 13 | \( 1 - 0.666T + 13T^{2} \) |
| 17 | \( 1 + 4.42T + 17T^{2} \) |
| 19 | \( 1 - 6.64T + 19T^{2} \) |
| 23 | \( 1 - 1.40T + 23T^{2} \) |
| 29 | \( 1 + 5.66T + 29T^{2} \) |
| 31 | \( 1 - 5.31T + 31T^{2} \) |
| 37 | \( 1 - 4.06T + 37T^{2} \) |
| 41 | \( 1 + 6.02T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 + 1.84T + 47T^{2} \) |
| 53 | \( 1 - 11.3T + 53T^{2} \) |
| 59 | \( 1 - 2.44T + 59T^{2} \) |
| 67 | \( 1 - 7.44T + 67T^{2} \) |
| 71 | \( 1 - 3.85T + 71T^{2} \) |
| 73 | \( 1 + 8.62T + 73T^{2} \) |
| 79 | \( 1 + 4.96T + 79T^{2} \) |
| 83 | \( 1 - 12.6T + 83T^{2} \) |
| 89 | \( 1 - 0.290T + 89T^{2} \) |
| 97 | \( 1 - 8.07T + 97T^{2} \) |
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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
−7.58238031052486293476866085382, −7.15047952731523010817235058771, −6.59705728095110357144703755454, −5.90999327081134964809014099800, −5.02062434535737617669696980531, −4.20549601859838907116653091659, −3.71237746917471256383924637597, −3.22471017617053536175832173128, −2.52332431453060942876476115270, −0.69852268814945840196459562004,
0.69852268814945840196459562004, 2.52332431453060942876476115270, 3.22471017617053536175832173128, 3.71237746917471256383924637597, 4.20549601859838907116653091659, 5.02062434535737617669696980531, 5.90999327081134964809014099800, 6.59705728095110357144703755454, 7.15047952731523010817235058771, 7.58238031052486293476866085382
