| L(s) = 1 | − 2.64·3-s + 5-s − 0.777·7-s + 4.01·9-s + 0.100·11-s + 5.88·13-s − 2.64·15-s − 4.46·17-s + 0.266·19-s + 2.05·21-s − 23-s + 25-s − 2.68·27-s − 1.56·29-s − 2.89·31-s − 0.266·33-s − 0.777·35-s + 1.38·37-s − 15.5·39-s − 0.958·41-s − 9.63·43-s + 4.01·45-s + 0.121·47-s − 6.39·49-s + 11.8·51-s + 9.63·53-s + 0.100·55-s + ⋯ |
| L(s) = 1 | − 1.52·3-s + 0.447·5-s − 0.293·7-s + 1.33·9-s + 0.0303·11-s + 1.63·13-s − 0.683·15-s − 1.08·17-s + 0.0611·19-s + 0.449·21-s − 0.208·23-s + 0.200·25-s − 0.517·27-s − 0.291·29-s − 0.520·31-s − 0.0463·33-s − 0.131·35-s + 0.227·37-s − 2.49·39-s − 0.149·41-s − 1.46·43-s + 0.598·45-s + 0.0177·47-s − 0.913·49-s + 1.65·51-s + 1.32·53-s + 0.0135·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + 2.64T + 3T^{2} \) |
| 7 | \( 1 + 0.777T + 7T^{2} \) |
| 11 | \( 1 - 0.100T + 11T^{2} \) |
| 13 | \( 1 - 5.88T + 13T^{2} \) |
| 17 | \( 1 + 4.46T + 17T^{2} \) |
| 19 | \( 1 - 0.266T + 19T^{2} \) |
| 29 | \( 1 + 1.56T + 29T^{2} \) |
| 31 | \( 1 + 2.89T + 31T^{2} \) |
| 37 | \( 1 - 1.38T + 37T^{2} \) |
| 41 | \( 1 + 0.958T + 41T^{2} \) |
| 43 | \( 1 + 9.63T + 43T^{2} \) |
| 47 | \( 1 - 0.121T + 47T^{2} \) |
| 53 | \( 1 - 9.63T + 53T^{2} \) |
| 59 | \( 1 + 1.48T + 59T^{2} \) |
| 61 | \( 1 - 5.39T + 61T^{2} \) |
| 67 | \( 1 - 9.85T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + 2.46T + 73T^{2} \) |
| 79 | \( 1 + 5.57T + 79T^{2} \) |
| 83 | \( 1 + 9.43T + 83T^{2} \) |
| 89 | \( 1 + 5.80T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−7.21351103629344571042333807353, −6.65585952523104366190555204573, −6.08470708790335579719027386822, −5.65585782759326676164759995489, −4.86251340744102452840658868955, −4.11332855677236069979638587199, −3.27501552095503159276891870463, −1.99275219034639791725311177391, −1.11324387529382691914424397338, 0,
1.11324387529382691914424397338, 1.99275219034639791725311177391, 3.27501552095503159276891870463, 4.11332855677236069979638587199, 4.86251340744102452840658868955, 5.65585782759326676164759995489, 6.08470708790335579719027386822, 6.65585952523104366190555204573, 7.21351103629344571042333807353
