| L(s) = 1 | − 7·3-s + 13·5-s − 16·7-s + 22·9-s − 45·11-s − 61·13-s − 91·15-s − 102·17-s − 68·19-s + 112·21-s − 194·23-s + 44·25-s + 35·27-s + 29·29-s − 149·31-s + 315·33-s − 208·35-s − 400·37-s + 427·39-s + 280·41-s + 263·43-s + 286·45-s − 509·47-s − 87·49-s + 714·51-s + 605·53-s − 585·55-s + ⋯ |
| L(s) = 1 | − 1.34·3-s + 1.16·5-s − 0.863·7-s + 0.814·9-s − 1.23·11-s − 1.30·13-s − 1.56·15-s − 1.45·17-s − 0.821·19-s + 1.16·21-s − 1.75·23-s + 0.351·25-s + 0.249·27-s + 0.185·29-s − 0.863·31-s + 1.66·33-s − 1.00·35-s − 1.77·37-s + 1.75·39-s + 1.06·41-s + 0.932·43-s + 0.947·45-s − 1.57·47-s − 0.253·49-s + 1.96·51-s + 1.56·53-s − 1.43·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1856 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 29 | \( 1 - p T \) |
| good | 3 | \( 1 + 7 T + p^{3} T^{2} \) |
| 5 | \( 1 - 13 T + p^{3} T^{2} \) |
| 7 | \( 1 + 16 T + p^{3} T^{2} \) |
| 11 | \( 1 + 45 T + p^{3} T^{2} \) |
| 13 | \( 1 + 61 T + p^{3} T^{2} \) |
| 17 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 19 | \( 1 + 68 T + p^{3} T^{2} \) |
| 23 | \( 1 + 194 T + p^{3} T^{2} \) |
| 31 | \( 1 + 149 T + p^{3} T^{2} \) |
| 37 | \( 1 + 400 T + p^{3} T^{2} \) |
| 41 | \( 1 - 280 T + p^{3} T^{2} \) |
| 43 | \( 1 - 263 T + p^{3} T^{2} \) |
| 47 | \( 1 + 509 T + p^{3} T^{2} \) |
| 53 | \( 1 - 605 T + p^{3} T^{2} \) |
| 59 | \( 1 + 578 T + p^{3} T^{2} \) |
| 61 | \( 1 - 718 T + p^{3} T^{2} \) |
| 67 | \( 1 + 260 T + p^{3} T^{2} \) |
| 71 | \( 1 + 738 T + p^{3} T^{2} \) |
| 73 | \( 1 - 652 T + p^{3} T^{2} \) |
| 79 | \( 1 - 917 T + p^{3} T^{2} \) |
| 83 | \( 1 - 678 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1008 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1764 T + p^{3} T^{2} \) |
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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
−8.025417810235534177782706988453, −6.91790443165846695146659537877, −6.40923224522338199398662700050, −5.64575707789603546714428899488, −5.14037201446945329063283619306, −4.16870872831896446982436758049, −2.59973859217988199322172260845, −1.97109772370186725936697464255, 0, 0,
1.97109772370186725936697464255, 2.59973859217988199322172260845, 4.16870872831896446982436758049, 5.14037201446945329063283619306, 5.64575707789603546714428899488, 6.40923224522338199398662700050, 6.91790443165846695146659537877, 8.025417810235534177782706988453
