L(s) = 1 | + 0.594·3-s + 1.41·5-s + 0.584·7-s − 2.64·9-s − 3.84·11-s + 0.483·13-s + 0.842·15-s − 3.45·17-s + 5.63·19-s + 0.347·21-s + 4.53·23-s − 2.99·25-s − 3.35·27-s + 3.21·29-s − 6.31·31-s − 2.28·33-s + 0.826·35-s + 11.3·37-s + 0.287·39-s + 0.951·41-s + 6.44·43-s − 3.74·45-s − 4.02·47-s − 6.65·49-s − 2.05·51-s + 12.9·53-s − 5.44·55-s + ⋯ |
L(s) = 1 | + 0.343·3-s + 0.633·5-s + 0.220·7-s − 0.882·9-s − 1.15·11-s + 0.134·13-s + 0.217·15-s − 0.837·17-s + 1.29·19-s + 0.0758·21-s + 0.946·23-s − 0.599·25-s − 0.646·27-s + 0.596·29-s − 1.13·31-s − 0.398·33-s + 0.139·35-s + 1.86·37-s + 0.0461·39-s + 0.148·41-s + 0.982·43-s − 0.558·45-s − 0.587·47-s − 0.951·49-s − 0.287·51-s + 1.77·53-s − 0.734·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.166790809\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.166790809\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 751 | \( 1 - T \) |
good | 3 | \( 1 - 0.594T + 3T^{2} \) |
| 5 | \( 1 - 1.41T + 5T^{2} \) |
| 7 | \( 1 - 0.584T + 7T^{2} \) |
| 11 | \( 1 + 3.84T + 11T^{2} \) |
| 13 | \( 1 - 0.483T + 13T^{2} \) |
| 17 | \( 1 + 3.45T + 17T^{2} \) |
| 19 | \( 1 - 5.63T + 19T^{2} \) |
| 23 | \( 1 - 4.53T + 23T^{2} \) |
| 29 | \( 1 - 3.21T + 29T^{2} \) |
| 31 | \( 1 + 6.31T + 31T^{2} \) |
| 37 | \( 1 - 11.3T + 37T^{2} \) |
| 41 | \( 1 - 0.951T + 41T^{2} \) |
| 43 | \( 1 - 6.44T + 43T^{2} \) |
| 47 | \( 1 + 4.02T + 47T^{2} \) |
| 53 | \( 1 - 12.9T + 53T^{2} \) |
| 59 | \( 1 - 5.19T + 59T^{2} \) |
| 61 | \( 1 - 10.0T + 61T^{2} \) |
| 67 | \( 1 + 15.3T + 67T^{2} \) |
| 71 | \( 1 - 8.95T + 71T^{2} \) |
| 73 | \( 1 - 4.45T + 73T^{2} \) |
| 79 | \( 1 - 4.96T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 6.61T + 89T^{2} \) |
| 97 | \( 1 + 7.23T + 97T^{2} \) |
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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
−8.030572116109270124306313250515, −7.54010777019583469208256895025, −6.66695964824857780499541213481, −5.72645025594506811408619556846, −5.39995967764710030250535562527, −4.54594388019667558499459949181, −3.45768922455815426417466004661, −2.67068662168513257600679355665, −2.10231048643644162447819702041, −0.74125293978578190952963240048,
0.74125293978578190952963240048, 2.10231048643644162447819702041, 2.67068662168513257600679355665, 3.45768922455815426417466004661, 4.54594388019667558499459949181, 5.39995967764710030250535562527, 5.72645025594506811408619556846, 6.66695964824857780499541213481, 7.54010777019583469208256895025, 8.030572116109270124306313250515