| L(s) = 1 | + 1.27·2-s − 0.753·3-s − 0.369·4-s − 0.962·6-s − 3.52·7-s − 3.02·8-s − 2.43·9-s + 1.51·11-s + 0.278·12-s − 2.48·13-s − 4.50·14-s − 3.12·16-s + 1.10·17-s − 3.10·18-s − 2.58·19-s + 2.65·21-s + 1.92·22-s − 7.60·23-s + 2.28·24-s − 3.16·26-s + 4.09·27-s + 1.30·28-s − 2.90·29-s + 1.56·31-s + 2.05·32-s − 1.13·33-s + 1.41·34-s + ⋯ |
| L(s) = 1 | + 0.903·2-s − 0.435·3-s − 0.184·4-s − 0.393·6-s − 1.33·7-s − 1.06·8-s − 0.810·9-s + 0.455·11-s + 0.0803·12-s − 0.687·13-s − 1.20·14-s − 0.781·16-s + 0.268·17-s − 0.731·18-s − 0.593·19-s + 0.580·21-s + 0.411·22-s − 1.58·23-s + 0.465·24-s − 0.621·26-s + 0.788·27-s + 0.246·28-s − 0.540·29-s + 0.280·31-s + 0.364·32-s − 0.198·33-s + 0.242·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5482741455\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5482741455\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 241 | \( 1 - T \) |
| good | 2 | \( 1 - 1.27T + 2T^{2} \) |
| 3 | \( 1 + 0.753T + 3T^{2} \) |
| 7 | \( 1 + 3.52T + 7T^{2} \) |
| 11 | \( 1 - 1.51T + 11T^{2} \) |
| 13 | \( 1 + 2.48T + 13T^{2} \) |
| 17 | \( 1 - 1.10T + 17T^{2} \) |
| 19 | \( 1 + 2.58T + 19T^{2} \) |
| 23 | \( 1 + 7.60T + 23T^{2} \) |
| 29 | \( 1 + 2.90T + 29T^{2} \) |
| 31 | \( 1 - 1.56T + 31T^{2} \) |
| 37 | \( 1 + 6.94T + 37T^{2} \) |
| 41 | \( 1 + 5.70T + 41T^{2} \) |
| 43 | \( 1 - 2.41T + 43T^{2} \) |
| 47 | \( 1 + 1.26T + 47T^{2} \) |
| 53 | \( 1 + 6.47T + 53T^{2} \) |
| 59 | \( 1 + 2.80T + 59T^{2} \) |
| 61 | \( 1 - 12.7T + 61T^{2} \) |
| 67 | \( 1 + 11.3T + 67T^{2} \) |
| 71 | \( 1 - 2.62T + 71T^{2} \) |
| 73 | \( 1 + 4.62T + 73T^{2} \) |
| 79 | \( 1 - 2.35T + 79T^{2} \) |
| 83 | \( 1 - 7.27T + 83T^{2} \) |
| 89 | \( 1 - 8.21T + 89T^{2} \) |
| 97 | \( 1 + 10.0T + 97T^{2} \) |
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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
−8.140395273639452260649939813338, −7.04890874233621255019583097050, −6.33867151446040398308572789677, −5.95350514247086505242763891671, −5.25827669564219395930239907192, −4.45674344029243639328004627897, −3.62879597357927779090180279533, −3.11944505377670951495494384824, −2.13503617387548190482642731789, −0.32236428451206177921787147208,
0.32236428451206177921787147208, 2.13503617387548190482642731789, 3.11944505377670951495494384824, 3.62879597357927779090180279533, 4.45674344029243639328004627897, 5.25827669564219395930239907192, 5.95350514247086505242763891671, 6.33867151446040398308572789677, 7.04890874233621255019583097050, 8.140395273639452260649939813338
