| L(s) = 1 | + 1.93·3-s + 3.14·5-s − 3.83·7-s + 0.727·9-s − 4.86·11-s + 4.27·13-s + 6.07·15-s − 6.83·17-s − 3.54·19-s − 7.41·21-s + 5.84·23-s + 4.91·25-s − 4.38·27-s − 9.26·29-s − 6.84·31-s − 9.39·33-s − 12.0·35-s − 5.31·37-s + 8.25·39-s + 1.64·41-s + 1.15·43-s + 2.29·45-s − 5.03·47-s + 7.74·49-s − 13.1·51-s + 9.73·53-s − 15.3·55-s + ⋯ |
| L(s) = 1 | + 1.11·3-s + 1.40·5-s − 1.45·7-s + 0.242·9-s − 1.46·11-s + 1.18·13-s + 1.56·15-s − 1.65·17-s − 0.812·19-s − 1.61·21-s + 1.21·23-s + 0.982·25-s − 0.844·27-s − 1.72·29-s − 1.22·31-s − 1.63·33-s − 2.04·35-s − 0.874·37-s + 1.32·39-s + 0.256·41-s + 0.176·43-s + 0.341·45-s − 0.733·47-s + 1.10·49-s − 1.84·51-s + 1.33·53-s − 2.06·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 251 | \( 1 - T \) |
| good | 3 | \( 1 - 1.93T + 3T^{2} \) |
| 5 | \( 1 - 3.14T + 5T^{2} \) |
| 7 | \( 1 + 3.83T + 7T^{2} \) |
| 11 | \( 1 + 4.86T + 11T^{2} \) |
| 13 | \( 1 - 4.27T + 13T^{2} \) |
| 17 | \( 1 + 6.83T + 17T^{2} \) |
| 19 | \( 1 + 3.54T + 19T^{2} \) |
| 23 | \( 1 - 5.84T + 23T^{2} \) |
| 29 | \( 1 + 9.26T + 29T^{2} \) |
| 31 | \( 1 + 6.84T + 31T^{2} \) |
| 37 | \( 1 + 5.31T + 37T^{2} \) |
| 41 | \( 1 - 1.64T + 41T^{2} \) |
| 43 | \( 1 - 1.15T + 43T^{2} \) |
| 47 | \( 1 + 5.03T + 47T^{2} \) |
| 53 | \( 1 - 9.73T + 53T^{2} \) |
| 59 | \( 1 + 2.57T + 59T^{2} \) |
| 61 | \( 1 - 3.42T + 61T^{2} \) |
| 67 | \( 1 - 13.5T + 67T^{2} \) |
| 71 | \( 1 - 8.49T + 71T^{2} \) |
| 73 | \( 1 + 9.42T + 73T^{2} \) |
| 79 | \( 1 + 7.49T + 79T^{2} \) |
| 83 | \( 1 + 5.19T + 83T^{2} \) |
| 89 | \( 1 + 4.74T + 89T^{2} \) |
| 97 | \( 1 + 0.648T + 97T^{2} \) |
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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
−8.401453403806330443205023844714, −7.23917230359251475946504901256, −6.65182688353913490650267583765, −5.83576496477074861390564404591, −5.33618101082290491315410561866, −3.97474066980723725275634062996, −3.21113825749043607628375434575, −2.45482680997341606338609527631, −1.90335421010329602921815275212, 0,
1.90335421010329602921815275212, 2.45482680997341606338609527631, 3.21113825749043607628375434575, 3.97474066980723725275634062996, 5.33618101082290491315410561866, 5.83576496477074861390564404591, 6.65182688353913490650267583765, 7.23917230359251475946504901256, 8.401453403806330443205023844714
