| L(s) = 1 | + 0.791·3-s + 0.178·5-s + 0.0809·7-s − 2.37·9-s + 4.30·11-s + 1.10·13-s + 0.141·15-s − 3.06·17-s + 2.15·19-s + 0.0640·21-s − 9.15·23-s − 4.96·25-s − 4.25·27-s − 6.17·29-s + 9.82·31-s + 3.40·33-s + 0.0144·35-s + 4.08·37-s + 0.871·39-s + 4.58·41-s − 2.03·43-s − 0.424·45-s + 5.12·47-s − 6.99·49-s − 2.42·51-s + 2.69·53-s + 0.769·55-s + ⋯ |
| L(s) = 1 | + 0.457·3-s + 0.0799·5-s + 0.0305·7-s − 0.791·9-s + 1.29·11-s + 0.305·13-s + 0.0365·15-s − 0.743·17-s + 0.495·19-s + 0.0139·21-s − 1.90·23-s − 0.993·25-s − 0.818·27-s − 1.14·29-s + 1.76·31-s + 0.593·33-s + 0.00244·35-s + 0.671·37-s + 0.139·39-s + 0.716·41-s − 0.309·43-s − 0.0632·45-s + 0.748·47-s − 0.999·49-s − 0.339·51-s + 0.369·53-s + 0.103·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 0.791T + 3T^{2} \) |
| 5 | \( 1 - 0.178T + 5T^{2} \) |
| 7 | \( 1 - 0.0809T + 7T^{2} \) |
| 11 | \( 1 - 4.30T + 11T^{2} \) |
| 13 | \( 1 - 1.10T + 13T^{2} \) |
| 17 | \( 1 + 3.06T + 17T^{2} \) |
| 19 | \( 1 - 2.15T + 19T^{2} \) |
| 23 | \( 1 + 9.15T + 23T^{2} \) |
| 29 | \( 1 + 6.17T + 29T^{2} \) |
| 31 | \( 1 - 9.82T + 31T^{2} \) |
| 37 | \( 1 - 4.08T + 37T^{2} \) |
| 41 | \( 1 - 4.58T + 41T^{2} \) |
| 43 | \( 1 + 2.03T + 43T^{2} \) |
| 47 | \( 1 - 5.12T + 47T^{2} \) |
| 53 | \( 1 - 2.69T + 53T^{2} \) |
| 59 | \( 1 + 9.21T + 59T^{2} \) |
| 61 | \( 1 + 14.0T + 61T^{2} \) |
| 67 | \( 1 + 5.62T + 67T^{2} \) |
| 71 | \( 1 - 4.88T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 17.4T + 79T^{2} \) |
| 83 | \( 1 - 2.48T + 83T^{2} \) |
| 89 | \( 1 - 5.57T + 89T^{2} \) |
| 97 | \( 1 + 13.9T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.78000963423418708688024436398, −6.64743302440547111249649712811, −6.11839407206300306288846523488, −5.61423608221065638469053335887, −4.37562784754540117901587333437, −3.97813460461231242897749855162, −3.09704685421057847738343832529, −2.23379838047914627923596684579, −1.41187947184039973695053820193, 0,
1.41187947184039973695053820193, 2.23379838047914627923596684579, 3.09704685421057847738343832529, 3.97813460461231242897749855162, 4.37562784754540117901587333437, 5.61423608221065638469053335887, 6.11839407206300306288846523488, 6.64743302440547111249649712811, 7.78000963423418708688024436398
