L(s) = 1 | + 3-s + 5-s + 5·7-s + 9-s − 11-s + 13-s + 15-s − 3·17-s − 6·19-s + 5·21-s + 7·23-s + 25-s + 27-s + 6·29-s + 2·31-s − 33-s + 5·35-s + 37-s + 39-s + 7·41-s − 8·43-s + 45-s − 2·47-s + 18·49-s − 3·51-s + 13·53-s − 55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.88·7-s + 1/3·9-s − 0.301·11-s + 0.277·13-s + 0.258·15-s − 0.727·17-s − 1.37·19-s + 1.09·21-s + 1.45·23-s + 1/5·25-s + 0.192·27-s + 1.11·29-s + 0.359·31-s − 0.174·33-s + 0.845·35-s + 0.164·37-s + 0.160·39-s + 1.09·41-s − 1.21·43-s + 0.149·45-s − 0.291·47-s + 18/7·49-s − 0.420·51-s + 1.78·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3120 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.230708004\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.230708004\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
good | 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 + T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 7 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 13 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 7 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 8 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−8.636379724815656251291831833952, −8.149231200809546435178549270845, −7.29698869311759236941868561697, −6.53376032748021268901082628279, −5.51010032514464960489476741133, −4.68478635524710604875861039237, −4.24495811928688089388599255403, −2.81839254914111434082656098200, −2.07691413454105786395434301306, −1.16986704567745584856462262998,
1.16986704567745584856462262998, 2.07691413454105786395434301306, 2.81839254914111434082656098200, 4.24495811928688089388599255403, 4.68478635524710604875861039237, 5.51010032514464960489476741133, 6.53376032748021268901082628279, 7.29698869311759236941868561697, 8.149231200809546435178549270845, 8.636379724815656251291831833952