| L(s) = 1 | − 8.72·3-s + 5·5-s + 5.16·7-s + 49.1·9-s − 38.0·11-s + 39.8·13-s − 43.6·15-s − 52.1·17-s + 35.3·19-s − 45.0·21-s + 23·23-s + 25·25-s − 193.·27-s + 154.·29-s − 275.·31-s + 331.·33-s + 25.8·35-s − 315.·37-s − 347.·39-s + 195.·41-s − 158.·43-s + 245.·45-s + 358.·47-s − 316.·49-s + 455.·51-s + 50.4·53-s − 190.·55-s + ⋯ |
| L(s) = 1 | − 1.67·3-s + 0.447·5-s + 0.278·7-s + 1.82·9-s − 1.04·11-s + 0.850·13-s − 0.751·15-s − 0.744·17-s + 0.426·19-s − 0.468·21-s + 0.208·23-s + 0.200·25-s − 1.38·27-s + 0.988·29-s − 1.59·31-s + 1.75·33-s + 0.124·35-s − 1.40·37-s − 1.42·39-s + 0.744·41-s − 0.562·43-s + 0.814·45-s + 1.11·47-s − 0.922·49-s + 1.25·51-s + 0.130·53-s − 0.465·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
| good | 3 | \( 1 + 8.72T + 27T^{2} \) |
| 7 | \( 1 - 5.16T + 343T^{2} \) |
| 11 | \( 1 + 38.0T + 1.33e3T^{2} \) |
| 13 | \( 1 - 39.8T + 2.19e3T^{2} \) |
| 17 | \( 1 + 52.1T + 4.91e3T^{2} \) |
| 19 | \( 1 - 35.3T + 6.85e3T^{2} \) |
| 29 | \( 1 - 154.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 275.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 315.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 195.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 158.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 358.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 50.4T + 1.48e5T^{2} \) |
| 59 | \( 1 + 408.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 218.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 228.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 850.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 675.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 933.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 663.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 243.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 585.T + 9.12e5T^{2} \) |
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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
−8.509939931272652761467408599722, −7.49787879898458228388849350466, −6.69441064506753449746860711157, −6.01430378890785619014079496749, −5.25478621854143859432528327277, −4.80175773628930691825307024559, −3.58710645965383676301006644124, −2.14631043315500016280924519100, −1.05198036041504224483051805700, 0,
1.05198036041504224483051805700, 2.14631043315500016280924519100, 3.58710645965383676301006644124, 4.80175773628930691825307024559, 5.25478621854143859432528327277, 6.01430378890785619014079496749, 6.69441064506753449746860711157, 7.49787879898458228388849350466, 8.509939931272652761467408599722
