| L(s) = 1 | + 2.56·3-s − 2·5-s + 2.56·7-s + 3.56·9-s − 2.56·11-s − 2·13-s − 5.12·15-s + 7.12·17-s + 7.12·19-s + 6.56·21-s − 2·23-s − 25-s + 1.43·27-s + 8.24·29-s − 0.876·31-s − 6.56·33-s − 5.12·35-s + 37-s − 5.12·39-s + 4.56·41-s − 8.24·43-s − 7.12·45-s + 10.5·47-s − 0.438·49-s + 18.2·51-s + 6.80·53-s + 5.12·55-s + ⋯ |
| L(s) = 1 | + 1.47·3-s − 0.894·5-s + 0.968·7-s + 1.18·9-s − 0.772·11-s − 0.554·13-s − 1.32·15-s + 1.72·17-s + 1.63·19-s + 1.43·21-s − 0.417·23-s − 0.200·25-s + 0.276·27-s + 1.53·29-s − 0.157·31-s − 1.14·33-s − 0.865·35-s + 0.164·37-s − 0.820·39-s + 0.712·41-s − 1.25·43-s − 1.06·45-s + 1.54·47-s − 0.0626·49-s + 2.55·51-s + 0.935·53-s + 0.690·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2368 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.967673900\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.967673900\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 3 | \( 1 - 2.56T + 3T^{2} \) |
| 5 | \( 1 + 2T + 5T^{2} \) |
| 7 | \( 1 - 2.56T + 7T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 7.12T + 17T^{2} \) |
| 19 | \( 1 - 7.12T + 19T^{2} \) |
| 23 | \( 1 + 2T + 23T^{2} \) |
| 29 | \( 1 - 8.24T + 29T^{2} \) |
| 31 | \( 1 + 0.876T + 31T^{2} \) |
| 41 | \( 1 - 4.56T + 41T^{2} \) |
| 43 | \( 1 + 8.24T + 43T^{2} \) |
| 47 | \( 1 - 10.5T + 47T^{2} \) |
| 53 | \( 1 - 6.80T + 53T^{2} \) |
| 59 | \( 1 + 3.12T + 59T^{2} \) |
| 61 | \( 1 - 11.1T + 61T^{2} \) |
| 67 | \( 1 - 14.2T + 67T^{2} \) |
| 71 | \( 1 - 5.43T + 71T^{2} \) |
| 73 | \( 1 + 15.9T + 73T^{2} \) |
| 79 | \( 1 + 9.36T + 79T^{2} \) |
| 83 | \( 1 + 1.43T + 83T^{2} \) |
| 89 | \( 1 - 4.87T + 89T^{2} \) |
| 97 | \( 1 + 0.876T + 97T^{2} \) |
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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
−8.653080021481302128096033900726, −8.157216979194321543713003659744, −7.60405355310465817574289093803, −7.28376513560825203454317452361, −5.62677407129253200434746486944, −4.90156267622169788776409236853, −3.90738364043563510734287945149, −3.17460394067185616157209702491, −2.39864602199403832499723584802, −1.10525777214681629735004881969,
1.10525777214681629735004881969, 2.39864602199403832499723584802, 3.17460394067185616157209702491, 3.90738364043563510734287945149, 4.90156267622169788776409236853, 5.62677407129253200434746486944, 7.28376513560825203454317452361, 7.60405355310465817574289093803, 8.157216979194321543713003659744, 8.653080021481302128096033900726
