| L(s) = 1 | − 5-s + 1.41·7-s + 1.41·11-s − 6.24·13-s − 3.24·17-s + 7.24·19-s + 2.65·23-s + 25-s − 4.24·29-s − 10.0·31-s − 1.41·35-s + 10.4·37-s + 2.24·41-s + 1.75·43-s − 11.6·47-s − 5·49-s + 7.24·53-s − 1.41·55-s − 13.0·59-s + 61-s + 6.24·65-s + 11.6·67-s − 13.4·71-s − 4.24·73-s + 2.00·77-s − 4.41·79-s − 11.8·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.534·7-s + 0.426·11-s − 1.73·13-s − 0.786·17-s + 1.66·19-s + 0.553·23-s + 0.200·25-s − 0.787·29-s − 1.80·31-s − 0.239·35-s + 1.72·37-s + 0.350·41-s + 0.267·43-s − 1.70·47-s − 0.714·49-s + 0.994·53-s − 0.190·55-s − 1.70·59-s + 0.128·61-s + 0.774·65-s + 1.42·67-s − 1.59·71-s − 0.496·73-s + 0.227·77-s − 0.496·79-s − 1.29·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 7 | \( 1 - 1.41T + 7T^{2} \) |
| 11 | \( 1 - 1.41T + 11T^{2} \) |
| 13 | \( 1 + 6.24T + 13T^{2} \) |
| 17 | \( 1 + 3.24T + 17T^{2} \) |
| 19 | \( 1 - 7.24T + 19T^{2} \) |
| 23 | \( 1 - 2.65T + 23T^{2} \) |
| 29 | \( 1 + 4.24T + 29T^{2} \) |
| 31 | \( 1 + 10.0T + 31T^{2} \) |
| 37 | \( 1 - 10.4T + 37T^{2} \) |
| 41 | \( 1 - 2.24T + 41T^{2} \) |
| 43 | \( 1 - 1.75T + 43T^{2} \) |
| 47 | \( 1 + 11.6T + 47T^{2} \) |
| 53 | \( 1 - 7.24T + 53T^{2} \) |
| 59 | \( 1 + 13.0T + 59T^{2} \) |
| 61 | \( 1 - T + 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 + 13.4T + 71T^{2} \) |
| 73 | \( 1 + 4.24T + 73T^{2} \) |
| 79 | \( 1 + 4.41T + 79T^{2} \) |
| 83 | \( 1 + 11.8T + 83T^{2} \) |
| 89 | \( 1 - 6.24T + 89T^{2} \) |
| 97 | \( 1 + 0.485T + 97T^{2} \) |
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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
−7.65793883862047655222307330505, −7.54320184892762765998642914752, −6.74266268144829381683809848746, −5.65305359389551361233582103338, −4.97074978390343268210677647770, −4.34835667328426724551263202326, −3.35502869995589768088769685140, −2.47311437625107703132860925347, −1.40716325360956402793950982725, 0,
1.40716325360956402793950982725, 2.47311437625107703132860925347, 3.35502869995589768088769685140, 4.34835667328426724551263202326, 4.97074978390343268210677647770, 5.65305359389551361233582103338, 6.74266268144829381683809848746, 7.54320184892762765998642914752, 7.65793883862047655222307330505
