L(s) = 1 | + 0.239·2-s + 3-s − 1.94·4-s − 0.863·5-s + 0.239·6-s − 3.18·7-s − 0.942·8-s + 9-s − 0.206·10-s + 2.43·11-s − 1.94·12-s + 2.64·13-s − 0.760·14-s − 0.863·15-s + 3.66·16-s + 17-s + 0.239·18-s − 5.74·19-s + 1.67·20-s − 3.18·21-s + 0.583·22-s + 5.49·23-s − 0.942·24-s − 4.25·25-s + 0.633·26-s + 27-s + 6.17·28-s + ⋯ |
L(s) = 1 | + 0.169·2-s + 0.577·3-s − 0.971·4-s − 0.385·5-s + 0.0976·6-s − 1.20·7-s − 0.333·8-s + 0.333·9-s − 0.0652·10-s + 0.735·11-s − 0.560·12-s + 0.734·13-s − 0.203·14-s − 0.222·15-s + 0.915·16-s + 0.242·17-s + 0.0563·18-s − 1.31·19-s + 0.374·20-s − 0.694·21-s + 0.124·22-s + 1.14·23-s − 0.192·24-s − 0.851·25-s + 0.124·26-s + 0.192·27-s + 1.16·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4029 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 79 | \( 1 + T \) |
good | 2 | \( 1 - 0.239T + 2T^{2} \) |
| 5 | \( 1 + 0.863T + 5T^{2} \) |
| 7 | \( 1 + 3.18T + 7T^{2} \) |
| 11 | \( 1 - 2.43T + 11T^{2} \) |
| 13 | \( 1 - 2.64T + 13T^{2} \) |
| 19 | \( 1 + 5.74T + 19T^{2} \) |
| 23 | \( 1 - 5.49T + 23T^{2} \) |
| 29 | \( 1 - 1.07T + 29T^{2} \) |
| 31 | \( 1 - 1.96T + 31T^{2} \) |
| 37 | \( 1 - 2.75T + 37T^{2} \) |
| 41 | \( 1 - 7.09T + 41T^{2} \) |
| 43 | \( 1 + 10.2T + 43T^{2} \) |
| 47 | \( 1 + 3.04T + 47T^{2} \) |
| 53 | \( 1 - 0.934T + 53T^{2} \) |
| 59 | \( 1 + 2.02T + 59T^{2} \) |
| 61 | \( 1 + 6.53T + 61T^{2} \) |
| 67 | \( 1 + 7.65T + 67T^{2} \) |
| 71 | \( 1 + 13.3T + 71T^{2} \) |
| 73 | \( 1 - 9.05T + 73T^{2} \) |
| 83 | \( 1 + 3.01T + 83T^{2} \) |
| 89 | \( 1 - 1.39T + 89T^{2} \) |
| 97 | \( 1 - 11.3T + 97T^{2} \) |
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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
−8.284803966669373118663682019087, −7.42906872834382737112948841985, −6.44668096990337373801826653509, −6.05475467014627099182304337518, −4.84884636132251015225680987539, −4.06687723551304086336480459519, −3.55561089574828021476333694318, −2.79802290994465333741159950655, −1.30331074567632905340387884302, 0,
1.30331074567632905340387884302, 2.79802290994465333741159950655, 3.55561089574828021476333694318, 4.06687723551304086336480459519, 4.84884636132251015225680987539, 6.05475467014627099182304337518, 6.44668096990337373801826653509, 7.42906872834382737112948841985, 8.284803966669373118663682019087