| L(s) = 1 | + 2-s − 3-s + 4-s + 2.46·5-s − 6-s + 7-s + 8-s + 9-s + 2.46·10-s − 3.81·11-s − 12-s − 13-s + 14-s − 2.46·15-s + 16-s − 17-s + 18-s − 7.17·19-s + 2.46·20-s − 21-s − 3.81·22-s + 0.960·23-s − 24-s + 1.06·25-s − 26-s − 27-s + 28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.10·5-s − 0.408·6-s + 0.377·7-s + 0.353·8-s + 0.333·9-s + 0.778·10-s − 1.14·11-s − 0.288·12-s − 0.277·13-s + 0.267·14-s − 0.635·15-s + 0.250·16-s − 0.242·17-s + 0.235·18-s − 1.64·19-s + 0.550·20-s − 0.218·21-s − 0.812·22-s + 0.200·23-s − 0.204·24-s + 0.212·25-s − 0.196·26-s − 0.192·27-s + 0.188·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9282 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 3 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| good | 5 | \( 1 - 2.46T + 5T^{2} \) |
| 11 | \( 1 + 3.81T + 11T^{2} \) |
| 19 | \( 1 + 7.17T + 19T^{2} \) |
| 23 | \( 1 - 0.960T + 23T^{2} \) |
| 29 | \( 1 - 0.852T + 29T^{2} \) |
| 31 | \( 1 - 4.45T + 31T^{2} \) |
| 37 | \( 1 + 10.0T + 37T^{2} \) |
| 41 | \( 1 - 4.07T + 41T^{2} \) |
| 43 | \( 1 + 7.63T + 43T^{2} \) |
| 47 | \( 1 - 5.69T + 47T^{2} \) |
| 53 | \( 1 - 0.711T + 53T^{2} \) |
| 59 | \( 1 - 4.28T + 59T^{2} \) |
| 61 | \( 1 + 6.27T + 61T^{2} \) |
| 67 | \( 1 + 4.59T + 67T^{2} \) |
| 71 | \( 1 - 6.47T + 71T^{2} \) |
| 73 | \( 1 + 14.2T + 73T^{2} \) |
| 79 | \( 1 - 5.69T + 79T^{2} \) |
| 83 | \( 1 + 7.49T + 83T^{2} \) |
| 89 | \( 1 + 8.07T + 89T^{2} \) |
| 97 | \( 1 + 17.5T + 97T^{2} \) |
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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
−7.05676513787402815685265005896, −6.58418344913479974453218691679, −5.80362890147078871761107195688, −5.39317380100986356607836006584, −4.71911182605009537035862657280, −4.09872783471223282599696354608, −2.88025525421910638634469004981, −2.23960432697388239149832544736, −1.51466465900279529869566545715, 0,
1.51466465900279529869566545715, 2.23960432697388239149832544736, 2.88025525421910638634469004981, 4.09872783471223282599696354608, 4.71911182605009537035862657280, 5.39317380100986356607836006584, 5.80362890147078871761107195688, 6.58418344913479974453218691679, 7.05676513787402815685265005896
