| L(s) = 1 | − 2.72·3-s − 2.84·7-s + 4.43·9-s − 0.0452·11-s − 2.94·13-s − 5.55·17-s − 1.80·19-s + 7.75·21-s − 7.18·23-s − 3.91·27-s − 0.322·29-s − 4.41·31-s + 0.123·33-s + 9.37·37-s + 8.01·39-s − 1.05·41-s − 6.99·43-s + 0.598·47-s + 1.09·49-s + 15.1·51-s + 12.8·53-s + 4.91·57-s − 9.08·59-s − 9.46·61-s − 12.6·63-s − 1.04·67-s + 19.5·69-s + ⋯ |
| L(s) = 1 | − 1.57·3-s − 1.07·7-s + 1.47·9-s − 0.0136·11-s − 0.815·13-s − 1.34·17-s − 0.413·19-s + 1.69·21-s − 1.49·23-s − 0.753·27-s − 0.0599·29-s − 0.792·31-s + 0.0214·33-s + 1.54·37-s + 1.28·39-s − 0.164·41-s − 1.06·43-s + 0.0872·47-s + 0.156·49-s + 2.12·51-s + 1.75·53-s + 0.651·57-s − 1.18·59-s − 1.21·61-s − 1.59·63-s − 0.127·67-s + 2.35·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1975058954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1975058954\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 2.72T + 3T^{2} \) |
| 7 | \( 1 + 2.84T + 7T^{2} \) |
| 11 | \( 1 + 0.0452T + 11T^{2} \) |
| 13 | \( 1 + 2.94T + 13T^{2} \) |
| 17 | \( 1 + 5.55T + 17T^{2} \) |
| 19 | \( 1 + 1.80T + 19T^{2} \) |
| 23 | \( 1 + 7.18T + 23T^{2} \) |
| 29 | \( 1 + 0.322T + 29T^{2} \) |
| 31 | \( 1 + 4.41T + 31T^{2} \) |
| 37 | \( 1 - 9.37T + 37T^{2} \) |
| 41 | \( 1 + 1.05T + 41T^{2} \) |
| 43 | \( 1 + 6.99T + 43T^{2} \) |
| 47 | \( 1 - 0.598T + 47T^{2} \) |
| 53 | \( 1 - 12.8T + 53T^{2} \) |
| 59 | \( 1 + 9.08T + 59T^{2} \) |
| 61 | \( 1 + 9.46T + 61T^{2} \) |
| 67 | \( 1 + 1.04T + 67T^{2} \) |
| 71 | \( 1 + 13.7T + 71T^{2} \) |
| 73 | \( 1 + 15.1T + 73T^{2} \) |
| 79 | \( 1 + 3.47T + 79T^{2} \) |
| 83 | \( 1 + 17.9T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 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
−8.490141656921501016162716898659, −7.36823989226548189652891238641, −6.86118686145374315650038070742, −6.03595643351130006937028825753, −5.81024304754038116126229434358, −4.62631015152994684756395988747, −4.21372639464010315272012712241, −2.93929047057528468361837332670, −1.83675006813623762523617089562, −0.26263975685922825557014063265,
0.26263975685922825557014063265, 1.83675006813623762523617089562, 2.93929047057528468361837332670, 4.21372639464010315272012712241, 4.62631015152994684756395988747, 5.81024304754038116126229434358, 6.03595643351130006937028825753, 6.86118686145374315650038070742, 7.36823989226548189652891238641, 8.490141656921501016162716898659
