| L(s) = 1 | + 0.0680·3-s − 3.21·5-s − 1.55·7-s − 2.99·9-s + 13-s − 0.218·15-s − 0.772·17-s − 4.41·19-s − 0.105·21-s − 2.60·23-s + 5.33·25-s − 0.408·27-s − 5.31·29-s + 3.08·31-s + 4.99·35-s − 6.07·37-s + 0.0680·39-s − 1.07·41-s − 9.12·43-s + 9.62·45-s + 4.53·47-s − 4.58·49-s − 0.0526·51-s − 0.569·53-s − 0.300·57-s − 14.1·59-s − 9.62·61-s + ⋯ |
| L(s) = 1 | + 0.0392·3-s − 1.43·5-s − 0.587·7-s − 0.998·9-s + 0.277·13-s − 0.0564·15-s − 0.187·17-s − 1.01·19-s − 0.0230·21-s − 0.542·23-s + 1.06·25-s − 0.0785·27-s − 0.987·29-s + 0.554·31-s + 0.844·35-s − 0.998·37-s + 0.0108·39-s − 0.168·41-s − 1.39·43-s + 1.43·45-s + 0.661·47-s − 0.654·49-s − 0.00736·51-s − 0.0781·53-s − 0.0397·57-s − 1.84·59-s − 1.23·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3445030208\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3445030208\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 - 0.0680T + 3T^{2} \) |
| 5 | \( 1 + 3.21T + 5T^{2} \) |
| 7 | \( 1 + 1.55T + 7T^{2} \) |
| 17 | \( 1 + 0.772T + 17T^{2} \) |
| 19 | \( 1 + 4.41T + 19T^{2} \) |
| 23 | \( 1 + 2.60T + 23T^{2} \) |
| 29 | \( 1 + 5.31T + 29T^{2} \) |
| 31 | \( 1 - 3.08T + 31T^{2} \) |
| 37 | \( 1 + 6.07T + 37T^{2} \) |
| 41 | \( 1 + 1.07T + 41T^{2} \) |
| 43 | \( 1 + 9.12T + 43T^{2} \) |
| 47 | \( 1 - 4.53T + 47T^{2} \) |
| 53 | \( 1 + 0.569T + 53T^{2} \) |
| 59 | \( 1 + 14.1T + 59T^{2} \) |
| 61 | \( 1 + 9.62T + 61T^{2} \) |
| 67 | \( 1 - 6.78T + 67T^{2} \) |
| 71 | \( 1 + 9.41T + 71T^{2} \) |
| 73 | \( 1 + 4.46T + 73T^{2} \) |
| 79 | \( 1 - 7.16T + 79T^{2} \) |
| 83 | \( 1 - 6.33T + 83T^{2} \) |
| 89 | \( 1 - 12.8T + 89T^{2} \) |
| 97 | \( 1 + 2.55T + 97T^{2} \) |
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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
−8.018807926145694032788723853747, −7.50081494841069312132145364530, −6.55879140764832871337753580558, −6.09690224030668598591969448269, −5.08754081955910266500775825447, −4.29718844878741258615173314302, −3.54487942175754980084273795177, −3.04682281204455332748153762023, −1.88703317622881733243122309984, −0.28823929599770783613366882953,
0.28823929599770783613366882953, 1.88703317622881733243122309984, 3.04682281204455332748153762023, 3.54487942175754980084273795177, 4.29718844878741258615173314302, 5.08754081955910266500775825447, 6.09690224030668598591969448269, 6.55879140764832871337753580558, 7.50081494841069312132145364530, 8.018807926145694032788723853747
