| L(s) = 1 | − 3-s − 4.38·5-s + 0.798·7-s + 9-s − 0.995·11-s + 0.108·13-s + 4.38·15-s − 4.71·17-s + 2.17·19-s − 0.798·21-s − 0.209·23-s + 14.2·25-s − 27-s + 1.12·29-s + 6.28·31-s + 0.995·33-s − 3.50·35-s − 11.2·37-s − 0.108·39-s − 4.79·41-s + 4.40·43-s − 4.38·45-s + 0.227·47-s − 6.36·49-s + 4.71·51-s + 10.8·53-s + 4.36·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.96·5-s + 0.301·7-s + 0.333·9-s − 0.300·11-s + 0.0302·13-s + 1.13·15-s − 1.14·17-s + 0.498·19-s − 0.174·21-s − 0.0437·23-s + 2.84·25-s − 0.192·27-s + 0.208·29-s + 1.12·31-s + 0.173·33-s − 0.591·35-s − 1.84·37-s − 0.0174·39-s − 0.748·41-s + 0.671·43-s − 0.653·45-s + 0.0331·47-s − 0.908·49-s + 0.660·51-s + 1.48·53-s + 0.588·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 + T \) |
| 251 | \( 1 - T \) |
| good | 5 | \( 1 + 4.38T + 5T^{2} \) |
| 7 | \( 1 - 0.798T + 7T^{2} \) |
| 11 | \( 1 + 0.995T + 11T^{2} \) |
| 13 | \( 1 - 0.108T + 13T^{2} \) |
| 17 | \( 1 + 4.71T + 17T^{2} \) |
| 19 | \( 1 - 2.17T + 19T^{2} \) |
| 23 | \( 1 + 0.209T + 23T^{2} \) |
| 29 | \( 1 - 1.12T + 29T^{2} \) |
| 31 | \( 1 - 6.28T + 31T^{2} \) |
| 37 | \( 1 + 11.2T + 37T^{2} \) |
| 41 | \( 1 + 4.79T + 41T^{2} \) |
| 43 | \( 1 - 4.40T + 43T^{2} \) |
| 47 | \( 1 - 0.227T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 0.194T + 59T^{2} \) |
| 61 | \( 1 - 10.8T + 61T^{2} \) |
| 67 | \( 1 - 9.06T + 67T^{2} \) |
| 71 | \( 1 + 0.886T + 71T^{2} \) |
| 73 | \( 1 - 16.0T + 73T^{2} \) |
| 79 | \( 1 - 13.7T + 79T^{2} \) |
| 83 | \( 1 + 3.64T + 83T^{2} \) |
| 89 | \( 1 - 8.58T + 89T^{2} \) |
| 97 | \( 1 + 17.5T + 97T^{2} \) |
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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
−7.78833332090101919072492785064, −6.89154445747316514279951974362, −6.67725582793184438221414263385, −5.29948878429345757847119552535, −4.83948919179025744572071141385, −4.04755170984497084748884176216, −3.48474427254438971165918671347, −2.39650018444175591993763998021, −0.959162154080802461802004115772, 0,
0.959162154080802461802004115772, 2.39650018444175591993763998021, 3.48474427254438971165918671347, 4.04755170984497084748884176216, 4.83948919179025744572071141385, 5.29948878429345757847119552535, 6.67725582793184438221414263385, 6.89154445747316514279951974362, 7.78833332090101919072492785064
