| L(s) = 1 | + 0.414·3-s − 2.82·5-s + 1.58·7-s − 2.82·9-s − 5.24·11-s − 1.17·15-s − 0.171·17-s + 7.24·19-s + 0.656·21-s − 7.24·23-s + 3.00·25-s − 2.41·27-s − 2.65·29-s − 5.65·31-s − 2.17·33-s − 4.48·35-s + 9.48·37-s − 0.171·41-s − 10.0·43-s + 8.00·45-s + 6·47-s − 4.48·49-s − 0.0710·51-s + 2.82·53-s + 14.8·55-s + 2.99·57-s + 7.24·59-s + ⋯ |
| L(s) = 1 | + 0.239·3-s − 1.26·5-s + 0.599·7-s − 0.942·9-s − 1.58·11-s − 0.302·15-s − 0.0416·17-s + 1.66·19-s + 0.143·21-s − 1.51·23-s + 0.600·25-s − 0.464·27-s − 0.493·29-s − 1.01·31-s − 0.378·33-s − 0.758·35-s + 1.55·37-s − 0.0267·41-s − 1.53·43-s + 1.19·45-s + 0.875·47-s − 0.640·49-s − 0.00995·51-s + 0.388·53-s + 1.99·55-s + 0.397·57-s + 0.942·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5408 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8980517640\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8980517640\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 0.414T + 3T^{2} \) |
| 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 - 1.58T + 7T^{2} \) |
| 11 | \( 1 + 5.24T + 11T^{2} \) |
| 17 | \( 1 + 0.171T + 17T^{2} \) |
| 19 | \( 1 - 7.24T + 19T^{2} \) |
| 23 | \( 1 + 7.24T + 23T^{2} \) |
| 29 | \( 1 + 2.65T + 29T^{2} \) |
| 31 | \( 1 + 5.65T + 31T^{2} \) |
| 37 | \( 1 - 9.48T + 37T^{2} \) |
| 41 | \( 1 + 0.171T + 41T^{2} \) |
| 43 | \( 1 + 10.0T + 43T^{2} \) |
| 47 | \( 1 - 6T + 47T^{2} \) |
| 53 | \( 1 - 2.82T + 53T^{2} \) |
| 59 | \( 1 - 7.24T + 59T^{2} \) |
| 61 | \( 1 - 7T + 61T^{2} \) |
| 67 | \( 1 - 4.75T + 67T^{2} \) |
| 71 | \( 1 - 1.24T + 71T^{2} \) |
| 73 | \( 1 + 4.48T + 73T^{2} \) |
| 79 | \( 1 + 6T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 14.6T + 89T^{2} \) |
| 97 | \( 1 + 9T + 97T^{2} \) |
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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
−8.018940152901382145738569340880, −7.74899505596805586378443603872, −7.05112330546481534770556673989, −5.70313849210100602167971837618, −5.39443459808881737353947382543, −4.45387158450239639287965451044, −3.61668044150343897118175462775, −2.93760772170531985918527862877, −2.02622705927015647123348692929, −0.47808060415157618940497261117,
0.47808060415157618940497261117, 2.02622705927015647123348692929, 2.93760772170531985918527862877, 3.61668044150343897118175462775, 4.45387158450239639287965451044, 5.39443459808881737353947382543, 5.70313849210100602167971837618, 7.05112330546481534770556673989, 7.74899505596805586378443603872, 8.018940152901382145738569340880
