| L(s) = 1 | − 3-s + 5-s + 4.60·7-s + 9-s + 11-s − 4.60·13-s − 15-s + 6.60·17-s − 7.21·19-s − 4.60·21-s + 25-s − 27-s + 8·29-s + 9.21·31-s − 33-s + 4.60·35-s − 3.21·37-s + 4.60·39-s + 8·41-s − 3.39·43-s + 45-s − 5.21·47-s + 14.2·49-s − 6.60·51-s + 2·53-s + 55-s + 7.21·57-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.447·5-s + 1.74·7-s + 0.333·9-s + 0.301·11-s − 1.27·13-s − 0.258·15-s + 1.60·17-s − 1.65·19-s − 1.00·21-s + 0.200·25-s − 0.192·27-s + 1.48·29-s + 1.65·31-s − 0.174·33-s + 0.778·35-s − 0.527·37-s + 0.737·39-s + 1.24·41-s − 0.517·43-s + 0.149·45-s − 0.760·47-s + 2.03·49-s − 0.924·51-s + 0.274·53-s + 0.134·55-s + 0.955·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 660 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 660 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.608497483\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.608497483\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| good | 7 | \( 1 - 4.60T + 7T^{2} \) |
| 13 | \( 1 + 4.60T + 13T^{2} \) |
| 17 | \( 1 - 6.60T + 17T^{2} \) |
| 19 | \( 1 + 7.21T + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 8T + 29T^{2} \) |
| 31 | \( 1 - 9.21T + 31T^{2} \) |
| 37 | \( 1 + 3.21T + 37T^{2} \) |
| 41 | \( 1 - 8T + 41T^{2} \) |
| 43 | \( 1 + 3.39T + 43T^{2} \) |
| 47 | \( 1 + 5.21T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 - 8T + 59T^{2} \) |
| 61 | \( 1 - 7.21T + 61T^{2} \) |
| 67 | \( 1 + 4T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 - 0.605T + 73T^{2} \) |
| 79 | \( 1 + 11.2T + 79T^{2} \) |
| 83 | \( 1 + 10.6T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 + 12.4T + 97T^{2} \) |
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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
−10.37631403991110681227434958537, −10.03021810875603782722581493037, −8.631126164531151275795714780150, −7.968445159272757478837738916360, −6.98926022453990045871448951669, −5.91546859152013761914224244641, −4.97062700541005579419933010852, −4.38058532201063213363452658859, −2.50369511805363283295151085528, −1.26358888529032209437631311984,
1.26358888529032209437631311984, 2.50369511805363283295151085528, 4.38058532201063213363452658859, 4.97062700541005579419933010852, 5.91546859152013761914224244641, 6.98926022453990045871448951669, 7.968445159272757478837738916360, 8.631126164531151275795714780150, 10.03021810875603782722581493037, 10.37631403991110681227434958537
