| L(s) = 1 | − 2.37·2-s + 1.95·3-s + 3.66·4-s − 4.66·6-s − 2.28·7-s − 3.95·8-s + 0.840·9-s + 1.12·11-s + 7.17·12-s − 5.95·13-s + 5.43·14-s + 2.09·16-s − 5.80·17-s − 2.00·18-s − 4.08·19-s − 4.47·21-s − 2.67·22-s + 23-s − 7.75·24-s + 14.1·26-s − 4.23·27-s − 8.36·28-s − 0.408·29-s − 3.19·31-s + 2.93·32-s + 2.20·33-s + 13.8·34-s + ⋯ |
| L(s) = 1 | − 1.68·2-s + 1.13·3-s + 1.83·4-s − 1.90·6-s − 0.863·7-s − 1.39·8-s + 0.280·9-s + 0.338·11-s + 2.07·12-s − 1.65·13-s + 1.45·14-s + 0.523·16-s − 1.40·17-s − 0.471·18-s − 0.936·19-s − 0.976·21-s − 0.570·22-s + 0.208·23-s − 1.58·24-s + 2.78·26-s − 0.814·27-s − 1.58·28-s − 0.0758·29-s − 0.573·31-s + 0.518·32-s + 0.383·33-s + 2.36·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 \) |
| 23 | \( 1 - T \) |
| good | 2 | \( 1 + 2.37T + 2T^{2} \) |
| 3 | \( 1 - 1.95T + 3T^{2} \) |
| 7 | \( 1 + 2.28T + 7T^{2} \) |
| 11 | \( 1 - 1.12T + 11T^{2} \) |
| 13 | \( 1 + 5.95T + 13T^{2} \) |
| 17 | \( 1 + 5.80T + 17T^{2} \) |
| 19 | \( 1 + 4.08T + 19T^{2} \) |
| 29 | \( 1 + 0.408T + 29T^{2} \) |
| 31 | \( 1 + 3.19T + 31T^{2} \) |
| 37 | \( 1 - 9.80T + 37T^{2} \) |
| 41 | \( 1 - 6.27T + 41T^{2} \) |
| 43 | \( 1 + 7.75T + 43T^{2} \) |
| 47 | \( 1 - 6.40T + 47T^{2} \) |
| 53 | \( 1 - 6.73T + 53T^{2} \) |
| 59 | \( 1 + 4.75T + 59T^{2} \) |
| 61 | \( 1 + 6.33T + 61T^{2} \) |
| 67 | \( 1 + 0.283T + 67T^{2} \) |
| 71 | \( 1 + 13.9T + 71T^{2} \) |
| 73 | \( 1 + 9.61T + 73T^{2} \) |
| 79 | \( 1 - 4.48T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 - 5.68T + 89T^{2} \) |
| 97 | \( 1 + 11.0T + 97T^{2} \) |
| 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
−9.889556754075627797029046287172, −9.223380231072993410758722877340, −8.835314815194349526028845914692, −7.81230632146078720605101239489, −7.13495112639324830089095307297, −6.24298680020580603624180141728, −4.36188884015730104378408567927, −2.81925552567233433081919613707, −2.11629666377104314734823351759, 0,
2.11629666377104314734823351759, 2.81925552567233433081919613707, 4.36188884015730104378408567927, 6.24298680020580603624180141728, 7.13495112639324830089095307297, 7.81230632146078720605101239489, 8.835314815194349526028845914692, 9.223380231072993410758722877340, 9.889556754075627797029046287172
