| L(s) = 1 | − 2.10·3-s − 1.81·5-s − 2.52·7-s + 1.42·9-s − 0.813·11-s − 4.33·13-s + 3.81·15-s − 0.916·17-s + 5.30·21-s − 1.81·23-s − 1.71·25-s + 3.31·27-s − 8.01·29-s − 9.30·31-s + 1.71·33-s + 4.57·35-s − 11.3·37-s + 9.12·39-s − 11.2·41-s + 9.96·43-s − 2.57·45-s − 5.64·47-s − 0.627·49-s + 1.92·51-s + 2.91·53-s + 1.47·55-s + 1.89·59-s + ⋯ |
| L(s) = 1 | − 1.21·3-s − 0.811·5-s − 0.954·7-s + 0.473·9-s − 0.245·11-s − 1.20·13-s + 0.984·15-s − 0.222·17-s + 1.15·21-s − 0.378·23-s − 0.342·25-s + 0.638·27-s − 1.48·29-s − 1.67·31-s + 0.297·33-s + 0.773·35-s − 1.85·37-s + 1.46·39-s − 1.75·41-s + 1.51·43-s − 0.384·45-s − 0.823·47-s − 0.0896·49-s + 0.269·51-s + 0.400·53-s + 0.198·55-s + 0.246·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2888 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.06866563932\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.06866563932\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 + 2.10T + 3T^{2} \) |
| 5 | \( 1 + 1.81T + 5T^{2} \) |
| 7 | \( 1 + 2.52T + 7T^{2} \) |
| 11 | \( 1 + 0.813T + 11T^{2} \) |
| 13 | \( 1 + 4.33T + 13T^{2} \) |
| 17 | \( 1 + 0.916T + 17T^{2} \) |
| 23 | \( 1 + 1.81T + 23T^{2} \) |
| 29 | \( 1 + 8.01T + 29T^{2} \) |
| 31 | \( 1 + 9.30T + 31T^{2} \) |
| 37 | \( 1 + 11.3T + 37T^{2} \) |
| 41 | \( 1 + 11.2T + 41T^{2} \) |
| 43 | \( 1 - 9.96T + 43T^{2} \) |
| 47 | \( 1 + 5.64T + 47T^{2} \) |
| 53 | \( 1 - 2.91T + 53T^{2} \) |
| 59 | \( 1 - 1.89T + 59T^{2} \) |
| 61 | \( 1 - 6.59T + 61T^{2} \) |
| 67 | \( 1 - 3.15T + 67T^{2} \) |
| 71 | \( 1 + 0.710T + 71T^{2} \) |
| 73 | \( 1 - 0.627T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 9.86T + 83T^{2} \) |
| 89 | \( 1 - 13.5T + 89T^{2} \) |
| 97 | \( 1 - 4.62T + 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.845845186564217396328418247166, −7.79578444146687154633985290496, −7.13010052319320668667941580425, −6.56627651566096033756357975462, −5.53936822567658684887672069347, −5.15481359527628758950317303555, −4.02530999882175080464096884737, −3.30930387187050434400058877868, −2.01346908494270965761317086639, −0.15983361109685320971249121864,
0.15983361109685320971249121864, 2.01346908494270965761317086639, 3.30930387187050434400058877868, 4.02530999882175080464096884737, 5.15481359527628758950317303555, 5.53936822567658684887672069347, 6.56627651566096033756357975462, 7.13010052319320668667941580425, 7.79578444146687154633985290496, 8.845845186564217396328418247166
