| L(s) = 1 | + 2-s + 2.38·3-s + 4-s + 3.46·5-s + 2.38·6-s + 3.50·7-s + 8-s + 2.69·9-s + 3.46·10-s + 4.98·11-s + 2.38·12-s − 6.89·13-s + 3.50·14-s + 8.25·15-s + 16-s + 0.889·17-s + 2.69·18-s + 19-s + 3.46·20-s + 8.36·21-s + 4.98·22-s − 4.08·23-s + 2.38·24-s + 6.97·25-s − 6.89·26-s − 0.724·27-s + 3.50·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.37·3-s + 0.5·4-s + 1.54·5-s + 0.974·6-s + 1.32·7-s + 0.353·8-s + 0.898·9-s + 1.09·10-s + 1.50·11-s + 0.688·12-s − 1.91·13-s + 0.936·14-s + 2.13·15-s + 0.250·16-s + 0.215·17-s + 0.635·18-s + 0.229·19-s + 0.773·20-s + 1.82·21-s + 1.06·22-s − 0.851·23-s + 0.487·24-s + 1.39·25-s − 1.35·26-s − 0.139·27-s + 0.662·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.799890522\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.799890522\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 19 | \( 1 - T \) |
| 211 | \( 1 - T \) |
| good | 3 | \( 1 - 2.38T + 3T^{2} \) |
| 5 | \( 1 - 3.46T + 5T^{2} \) |
| 7 | \( 1 - 3.50T + 7T^{2} \) |
| 11 | \( 1 - 4.98T + 11T^{2} \) |
| 13 | \( 1 + 6.89T + 13T^{2} \) |
| 17 | \( 1 - 0.889T + 17T^{2} \) |
| 23 | \( 1 + 4.08T + 23T^{2} \) |
| 29 | \( 1 + 2.32T + 29T^{2} \) |
| 31 | \( 1 + 2.78T + 31T^{2} \) |
| 37 | \( 1 - 8.46T + 37T^{2} \) |
| 41 | \( 1 + 10.3T + 41T^{2} \) |
| 43 | \( 1 + 12.2T + 43T^{2} \) |
| 47 | \( 1 + 3.48T + 47T^{2} \) |
| 53 | \( 1 - 5.29T + 53T^{2} \) |
| 59 | \( 1 + 2.41T + 59T^{2} \) |
| 61 | \( 1 + 6.90T + 61T^{2} \) |
| 67 | \( 1 - 15.3T + 67T^{2} \) |
| 71 | \( 1 - 9.85T + 71T^{2} \) |
| 73 | \( 1 - 14.3T + 73T^{2} \) |
| 79 | \( 1 + 3.20T + 79T^{2} \) |
| 83 | \( 1 + 6.78T + 83T^{2} \) |
| 89 | \( 1 + 18.1T + 89T^{2} \) |
| 97 | \( 1 + 5.28T + 97T^{2} \) |
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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
−7.924956321756959013844714087088, −7.07118279439488087649026863201, −6.53591769967873649085540282570, −5.52687153009703811758206071589, −5.04653401565160471265798427636, −4.28300061401685277837930730684, −3.45302755546449630877074241977, −2.52032058295597317287677602522, −1.93915804582939511816960290050, −1.52016706375611341583996453547,
1.52016706375611341583996453547, 1.93915804582939511816960290050, 2.52032058295597317287677602522, 3.45302755546449630877074241977, 4.28300061401685277837930730684, 5.04653401565160471265798427636, 5.52687153009703811758206071589, 6.53591769967873649085540282570, 7.07118279439488087649026863201, 7.924956321756959013844714087088
