| L(s) = 1 | − 1.79·2-s + 1.22·4-s + 4.13·5-s + 4.22·7-s + 1.38·8-s − 7.42·10-s − 11-s − 5.95·13-s − 7.59·14-s − 4.94·16-s − 6.87·17-s − 0.114·19-s + 5.07·20-s + 1.79·22-s − 3.90·23-s + 12.0·25-s + 10.6·26-s + 5.19·28-s − 1.80·29-s − 8.12·31-s + 6.12·32-s + 12.3·34-s + 17.4·35-s − 4.76·37-s + 0.205·38-s + 5.72·40-s + 1.81·41-s + ⋯ |
| L(s) = 1 | − 1.27·2-s + 0.614·4-s + 1.84·5-s + 1.59·7-s + 0.489·8-s − 2.34·10-s − 0.301·11-s − 1.65·13-s − 2.03·14-s − 1.23·16-s − 1.66·17-s − 0.0262·19-s + 1.13·20-s + 0.383·22-s − 0.814·23-s + 2.41·25-s + 2.09·26-s + 0.982·28-s − 0.335·29-s − 1.46·31-s + 1.08·32-s + 2.11·34-s + 2.95·35-s − 0.784·37-s + 0.0333·38-s + 0.904·40-s + 0.283·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6039 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| good | 2 | \( 1 + 1.79T + 2T^{2} \) |
| 5 | \( 1 - 4.13T + 5T^{2} \) |
| 7 | \( 1 - 4.22T + 7T^{2} \) |
| 13 | \( 1 + 5.95T + 13T^{2} \) |
| 17 | \( 1 + 6.87T + 17T^{2} \) |
| 19 | \( 1 + 0.114T + 19T^{2} \) |
| 23 | \( 1 + 3.90T + 23T^{2} \) |
| 29 | \( 1 + 1.80T + 29T^{2} \) |
| 31 | \( 1 + 8.12T + 31T^{2} \) |
| 37 | \( 1 + 4.76T + 37T^{2} \) |
| 41 | \( 1 - 1.81T + 41T^{2} \) |
| 43 | \( 1 - 0.000598T + 43T^{2} \) |
| 47 | \( 1 - 6.87T + 47T^{2} \) |
| 53 | \( 1 + 10.2T + 53T^{2} \) |
| 59 | \( 1 + 6.58T + 59T^{2} \) |
| 67 | \( 1 + 9.06T + 67T^{2} \) |
| 71 | \( 1 + 12.7T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 8.35T + 79T^{2} \) |
| 83 | \( 1 - 2.71T + 83T^{2} \) |
| 89 | \( 1 + 16.6T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−7.75530278803685248533095153232, −7.28846569767728412783184363284, −6.47779915715136782640843826541, −5.51651594685277554310636726129, −4.93839495674941282363225258148, −4.38800219107509387376919352313, −2.50472357572721534009471032571, −2.00325634024839314370607979298, −1.54047701720809678432649394306, 0,
1.54047701720809678432649394306, 2.00325634024839314370607979298, 2.50472357572721534009471032571, 4.38800219107509387376919352313, 4.93839495674941282363225258148, 5.51651594685277554310636726129, 6.47779915715136782640843826541, 7.28846569767728412783184363284, 7.75530278803685248533095153232
