| L(s) = 1 | + 0.585·5-s − 2.82·7-s + 11-s − 13-s − 2.24·17-s − 0.828·23-s − 4.65·25-s + 3.41·29-s − 2.58·31-s − 1.65·35-s − 4.82·37-s + 7.65·41-s + 8.24·43-s + 8·47-s + 1.00·49-s + 13.3·53-s + 0.585·55-s + 2.82·59-s − 8.82·61-s − 0.585·65-s − 4.24·67-s + 5.17·71-s + 5.65·73-s − 2.82·77-s + 7.07·79-s + 6.34·83-s − 1.31·85-s + ⋯ |
| L(s) = 1 | + 0.261·5-s − 1.06·7-s + 0.301·11-s − 0.277·13-s − 0.543·17-s − 0.172·23-s − 0.931·25-s + 0.634·29-s − 0.464·31-s − 0.280·35-s − 0.793·37-s + 1.19·41-s + 1.25·43-s + 1.16·47-s + 0.142·49-s + 1.82·53-s + 0.0789·55-s + 0.368·59-s − 1.13·61-s − 0.0726·65-s − 0.518·67-s + 0.613·71-s + 0.662·73-s − 0.322·77-s + 0.795·79-s + 0.696·83-s − 0.142·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5148 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.477586978\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.477586978\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 5 | \( 1 - 0.585T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 17 | \( 1 + 2.24T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 0.828T + 23T^{2} \) |
| 29 | \( 1 - 3.41T + 29T^{2} \) |
| 31 | \( 1 + 2.58T + 31T^{2} \) |
| 37 | \( 1 + 4.82T + 37T^{2} \) |
| 41 | \( 1 - 7.65T + 41T^{2} \) |
| 43 | \( 1 - 8.24T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 13.3T + 53T^{2} \) |
| 59 | \( 1 - 2.82T + 59T^{2} \) |
| 61 | \( 1 + 8.82T + 61T^{2} \) |
| 67 | \( 1 + 4.24T + 67T^{2} \) |
| 71 | \( 1 - 5.17T + 71T^{2} \) |
| 73 | \( 1 - 5.65T + 73T^{2} \) |
| 79 | \( 1 - 7.07T + 79T^{2} \) |
| 83 | \( 1 - 6.34T + 83T^{2} \) |
| 89 | \( 1 + 7.89T + 89T^{2} \) |
| 97 | \( 1 + 7.17T + 97T^{2} \) |
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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
−8.234660224075325914977177560030, −7.36922841768506823465703171028, −6.78916243371043227894085485476, −6.03847467339539931474354744464, −5.50803623186877780805906419898, −4.38469311310473439579898341168, −3.76417608090193862510810529222, −2.80068148331913175492017637877, −2.02491314588121081086746552796, −0.63994563114220351788982401690,
0.63994563114220351788982401690, 2.02491314588121081086746552796, 2.80068148331913175492017637877, 3.76417608090193862510810529222, 4.38469311310473439579898341168, 5.50803623186877780805906419898, 6.03847467339539931474354744464, 6.78916243371043227894085485476, 7.36922841768506823465703171028, 8.234660224075325914977177560030
