| L(s) = 1 | − 2.47·3-s − 4.14·5-s − 1.96·7-s + 3.14·9-s − 3.34·11-s − 13-s + 10.2·15-s − 4.86·17-s + 7.27·19-s + 4.86·21-s + 4.95·23-s + 12.1·25-s − 0.350·27-s − 2·29-s + 4.44·31-s + 8.28·33-s + 8.13·35-s − 2.58·37-s + 2.47·39-s + 11.7·41-s − 0.350·43-s − 13.0·45-s − 4.79·47-s − 3.14·49-s + 12.0·51-s − 13.7·53-s + 13.8·55-s + ⋯ |
| L(s) = 1 | − 1.43·3-s − 1.85·5-s − 0.742·7-s + 1.04·9-s − 1.00·11-s − 0.277·13-s + 2.64·15-s − 1.18·17-s + 1.66·19-s + 1.06·21-s + 1.03·23-s + 2.43·25-s − 0.0674·27-s − 0.371·29-s + 0.797·31-s + 1.44·33-s + 1.37·35-s − 0.425·37-s + 0.396·39-s + 1.83·41-s − 0.0534·43-s − 1.93·45-s − 0.699·47-s − 0.448·49-s + 1.68·51-s − 1.88·53-s + 1.86·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3328 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 2.47T + 3T^{2} \) |
| 5 | \( 1 + 4.14T + 5T^{2} \) |
| 7 | \( 1 + 1.96T + 7T^{2} \) |
| 11 | \( 1 + 3.34T + 11T^{2} \) |
| 17 | \( 1 + 4.86T + 17T^{2} \) |
| 19 | \( 1 - 7.27T + 19T^{2} \) |
| 23 | \( 1 - 4.95T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 4.44T + 31T^{2} \) |
| 37 | \( 1 + 2.58T + 37T^{2} \) |
| 41 | \( 1 - 11.7T + 41T^{2} \) |
| 43 | \( 1 + 0.350T + 43T^{2} \) |
| 47 | \( 1 + 4.79T + 47T^{2} \) |
| 53 | \( 1 + 13.7T + 53T^{2} \) |
| 59 | \( 1 - 2.64T + 59T^{2} \) |
| 61 | \( 1 - 1.45T + 61T^{2} \) |
| 67 | \( 1 + 5.54T + 67T^{2} \) |
| 71 | \( 1 - 0.936T + 71T^{2} \) |
| 73 | \( 1 - 0.829T + 73T^{2} \) |
| 79 | \( 1 - 11.6T + 79T^{2} \) |
| 83 | \( 1 - 8.99T + 83T^{2} \) |
| 89 | \( 1 + 6.28T + 89T^{2} \) |
| 97 | \( 1 - 14.5T + 97T^{2} \) |
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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
−7.996798643340687239016355570067, −7.41631382394924290144225248787, −6.81966700442792371812441023007, −6.03968786788623127017521439897, −4.92772177489552066458204457803, −4.73376431266844026150937210971, −3.54814959194508732384647777827, −2.82933111945574545674260778628, −0.810254849719169138051702764784, 0,
0.810254849719169138051702764784, 2.82933111945574545674260778628, 3.54814959194508732384647777827, 4.73376431266844026150937210971, 4.92772177489552066458204457803, 6.03968786788623127017521439897, 6.81966700442792371812441023007, 7.41631382394924290144225248787, 7.996798643340687239016355570067
