L(s) = 1 | + 2.42·2-s − 1.43·3-s + 3.85·4-s − 0.00337·5-s − 3.46·6-s − 4.65·7-s + 4.50·8-s − 0.945·9-s − 0.00815·10-s − 11-s − 5.53·12-s − 3.09·13-s − 11.2·14-s + 0.00483·15-s + 3.17·16-s − 17-s − 2.28·18-s + 1.66·19-s − 0.0130·20-s + 6.66·21-s − 2.42·22-s + 0.653·23-s − 6.45·24-s − 4.99·25-s − 7.50·26-s + 5.65·27-s − 17.9·28-s + ⋯ |
L(s) = 1 | + 1.71·2-s − 0.827·3-s + 1.92·4-s − 0.00150·5-s − 1.41·6-s − 1.75·7-s + 1.59·8-s − 0.315·9-s − 0.00257·10-s − 0.301·11-s − 1.59·12-s − 0.859·13-s − 3.01·14-s + 0.00124·15-s + 0.794·16-s − 0.242·17-s − 0.539·18-s + 0.383·19-s − 0.00290·20-s + 1.45·21-s − 0.516·22-s + 0.136·23-s − 1.31·24-s − 0.999·25-s − 1.47·26-s + 1.08·27-s − 3.39·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8041 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.245259465\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.245259465\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 43 | \( 1 - T \) |
good | 2 | \( 1 - 2.42T + 2T^{2} \) |
| 3 | \( 1 + 1.43T + 3T^{2} \) |
| 5 | \( 1 + 0.00337T + 5T^{2} \) |
| 7 | \( 1 + 4.65T + 7T^{2} \) |
| 13 | \( 1 + 3.09T + 13T^{2} \) |
| 19 | \( 1 - 1.66T + 19T^{2} \) |
| 23 | \( 1 - 0.653T + 23T^{2} \) |
| 29 | \( 1 - 6.43T + 29T^{2} \) |
| 31 | \( 1 + 1.77T + 31T^{2} \) |
| 37 | \( 1 - 7.07T + 37T^{2} \) |
| 41 | \( 1 + 3.90T + 41T^{2} \) |
| 47 | \( 1 - 4.49T + 47T^{2} \) |
| 53 | \( 1 - 5.29T + 53T^{2} \) |
| 59 | \( 1 + 3.35T + 59T^{2} \) |
| 61 | \( 1 - 6.18T + 61T^{2} \) |
| 67 | \( 1 - 9.54T + 67T^{2} \) |
| 71 | \( 1 - 16.4T + 71T^{2} \) |
| 73 | \( 1 - 12.3T + 73T^{2} \) |
| 79 | \( 1 + 3.55T + 79T^{2} \) |
| 83 | \( 1 + 5.51T + 83T^{2} \) |
| 89 | \( 1 + 2.24T + 89T^{2} \) |
| 97 | \( 1 - 3.26T + 97T^{2} \) |
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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
−7.34327600088841197100910846201, −6.65670080691729985512708186905, −6.36797394552434766613890055964, −5.56018665057885561513046236782, −5.24677582931865240959030537961, −4.31744718064605403109478001166, −3.63017520840019953165056201941, −2.83450030981822942582472307455, −2.37641815781911052739944147667, −0.55069992796881480315675002948,
0.55069992796881480315675002948, 2.37641815781911052739944147667, 2.83450030981822942582472307455, 3.63017520840019953165056201941, 4.31744718064605403109478001166, 5.24677582931865240959030537961, 5.56018665057885561513046236782, 6.36797394552434766613890055964, 6.65670080691729985512708186905, 7.34327600088841197100910846201