| L(s) = 1 | + 0.652·3-s − 1.34·5-s − 1.53·7-s − 2.57·9-s + 1.18·11-s + 2.71·13-s − 0.879·15-s + 3.87·17-s − 0.999·21-s + 5.06·23-s − 3.18·25-s − 3.63·27-s − 4.65·29-s + 3.83·31-s + 0.773·33-s + 2.06·35-s − 4.10·37-s + 1.77·39-s − 9.98·41-s + 8.70·43-s + 3.46·45-s − 0.573·47-s − 4.65·49-s + 2.53·51-s + 2.94·53-s − 1.59·55-s + 3.93·59-s + ⋯ |
| L(s) = 1 | + 0.376·3-s − 0.602·5-s − 0.579·7-s − 0.857·9-s + 0.357·11-s + 0.753·13-s − 0.227·15-s + 0.940·17-s − 0.218·21-s + 1.05·23-s − 0.636·25-s − 0.700·27-s − 0.863·29-s + 0.689·31-s + 0.134·33-s + 0.348·35-s − 0.675·37-s + 0.283·39-s − 1.55·41-s + 1.32·43-s + 0.516·45-s − 0.0837·47-s − 0.664·49-s + 0.354·51-s + 0.404·53-s − 0.215·55-s + 0.512·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5776 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.593638958\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.593638958\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 0.652T + 3T^{2} \) |
| 5 | \( 1 + 1.34T + 5T^{2} \) |
| 7 | \( 1 + 1.53T + 7T^{2} \) |
| 11 | \( 1 - 1.18T + 11T^{2} \) |
| 13 | \( 1 - 2.71T + 13T^{2} \) |
| 17 | \( 1 - 3.87T + 17T^{2} \) |
| 23 | \( 1 - 5.06T + 23T^{2} \) |
| 29 | \( 1 + 4.65T + 29T^{2} \) |
| 31 | \( 1 - 3.83T + 31T^{2} \) |
| 37 | \( 1 + 4.10T + 37T^{2} \) |
| 41 | \( 1 + 9.98T + 41T^{2} \) |
| 43 | \( 1 - 8.70T + 43T^{2} \) |
| 47 | \( 1 + 0.573T + 47T^{2} \) |
| 53 | \( 1 - 2.94T + 53T^{2} \) |
| 59 | \( 1 - 3.93T + 59T^{2} \) |
| 61 | \( 1 + 4.51T + 61T^{2} \) |
| 67 | \( 1 - 3.88T + 67T^{2} \) |
| 71 | \( 1 - 6.93T + 71T^{2} \) |
| 73 | \( 1 - 6.12T + 73T^{2} \) |
| 79 | \( 1 + 9.80T + 79T^{2} \) |
| 83 | \( 1 + 12.3T + 83T^{2} \) |
| 89 | \( 1 + 2.42T + 89T^{2} \) |
| 97 | \( 1 - 7.36T + 97T^{2} \) |
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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
−8.222083949267612361059886243975, −7.45855918583571539022791024073, −6.76237520795418354202153034678, −5.93479497115190730285937439639, −5.36731784358716291234650699222, −4.28848903492114018552127155808, −3.40214595020986896323992559767, −3.15830212567872050310513168802, −1.89629061674838337880160299969, −0.64896565349857841168088544286,
0.64896565349857841168088544286, 1.89629061674838337880160299969, 3.15830212567872050310513168802, 3.40214595020986896323992559767, 4.28848903492114018552127155808, 5.36731784358716291234650699222, 5.93479497115190730285937439639, 6.76237520795418354202153034678, 7.45855918583571539022791024073, 8.222083949267612361059886243975
