| L(s) = 1 | − 3-s − 4.11·7-s + 9-s − 3.21·11-s − 13-s + 6.05·17-s − 2.90·19-s + 4.11·21-s + 3.65·23-s − 27-s − 0.377·29-s − 1.40·31-s + 3.21·33-s + 10.0·37-s + 39-s + 1.33·41-s + 6.57·43-s − 8.11·47-s + 9.95·49-s − 6.05·51-s + 4.09·53-s + 2.90·57-s − 14.2·59-s − 2.57·61-s − 4.11·63-s + 7.02·67-s − 3.65·69-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.55·7-s + 0.333·9-s − 0.969·11-s − 0.277·13-s + 1.46·17-s − 0.666·19-s + 0.898·21-s + 0.762·23-s − 0.192·27-s − 0.0701·29-s − 0.252·31-s + 0.559·33-s + 1.65·37-s + 0.160·39-s + 0.208·41-s + 1.00·43-s − 1.18·47-s + 1.42·49-s − 0.847·51-s + 0.562·53-s + 0.384·57-s − 1.85·59-s − 0.329·61-s − 0.518·63-s + 0.857·67-s − 0.440·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 7 | \( 1 + 4.11T + 7T^{2} \) |
| 11 | \( 1 + 3.21T + 11T^{2} \) |
| 17 | \( 1 - 6.05T + 17T^{2} \) |
| 19 | \( 1 + 2.90T + 19T^{2} \) |
| 23 | \( 1 - 3.65T + 23T^{2} \) |
| 29 | \( 1 + 0.377T + 29T^{2} \) |
| 31 | \( 1 + 1.40T + 31T^{2} \) |
| 37 | \( 1 - 10.0T + 37T^{2} \) |
| 41 | \( 1 - 1.33T + 41T^{2} \) |
| 43 | \( 1 - 6.57T + 43T^{2} \) |
| 47 | \( 1 + 8.11T + 47T^{2} \) |
| 53 | \( 1 - 4.09T + 53T^{2} \) |
| 59 | \( 1 + 14.2T + 59T^{2} \) |
| 61 | \( 1 + 2.57T + 61T^{2} \) |
| 67 | \( 1 - 7.02T + 67T^{2} \) |
| 71 | \( 1 - 7.19T + 71T^{2} \) |
| 73 | \( 1 - 7.46T + 73T^{2} \) |
| 79 | \( 1 - 7.13T + 79T^{2} \) |
| 83 | \( 1 - 10.9T + 83T^{2} \) |
| 89 | \( 1 + 18.5T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−7.53187686372427572615055812449, −6.67446114524337539152497094972, −6.14357637020982794866056435722, −5.51136156742285234112041417663, −4.81565279272108814951564343679, −3.84233490312395565961706473238, −3.09696383709189731672943090669, −2.42578313508821444493941527897, −0.988677189165820810713725797577, 0,
0.988677189165820810713725797577, 2.42578313508821444493941527897, 3.09696383709189731672943090669, 3.84233490312395565961706473238, 4.81565279272108814951564343679, 5.51136156742285234112041417663, 6.14357637020982794866056435722, 6.67446114524337539152497094972, 7.53187686372427572615055812449
