| L(s) = 1 | + 2.17·3-s + 2.95·5-s − 2.27·7-s + 1.72·9-s − 3.76·11-s − 4.15·13-s + 6.41·15-s + 4.07·17-s + 0.0709·19-s − 4.94·21-s + 1.46·23-s + 3.71·25-s − 2.78·27-s − 7.74·29-s − 4.27·31-s − 8.17·33-s − 6.71·35-s + 2.89·37-s − 9.03·39-s − 4.41·41-s − 5.20·43-s + 5.07·45-s + 3.09·47-s − 1.82·49-s + 8.86·51-s + 0.690·53-s − 11.1·55-s + ⋯ |
| L(s) = 1 | + 1.25·3-s + 1.32·5-s − 0.859·7-s + 0.573·9-s − 1.13·11-s − 1.15·13-s + 1.65·15-s + 0.989·17-s + 0.0162·19-s − 1.07·21-s + 0.306·23-s + 0.742·25-s − 0.535·27-s − 1.43·29-s − 0.767·31-s − 1.42·33-s − 1.13·35-s + 0.476·37-s − 1.44·39-s − 0.688·41-s − 0.793·43-s + 0.756·45-s + 0.451·47-s − 0.260·49-s + 1.24·51-s + 0.0948·53-s − 1.49·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 503 | \( 1 - T \) |
| good | 3 | \( 1 - 2.17T + 3T^{2} \) |
| 5 | \( 1 - 2.95T + 5T^{2} \) |
| 7 | \( 1 + 2.27T + 7T^{2} \) |
| 11 | \( 1 + 3.76T + 11T^{2} \) |
| 13 | \( 1 + 4.15T + 13T^{2} \) |
| 17 | \( 1 - 4.07T + 17T^{2} \) |
| 19 | \( 1 - 0.0709T + 19T^{2} \) |
| 23 | \( 1 - 1.46T + 23T^{2} \) |
| 29 | \( 1 + 7.74T + 29T^{2} \) |
| 31 | \( 1 + 4.27T + 31T^{2} \) |
| 37 | \( 1 - 2.89T + 37T^{2} \) |
| 41 | \( 1 + 4.41T + 41T^{2} \) |
| 43 | \( 1 + 5.20T + 43T^{2} \) |
| 47 | \( 1 - 3.09T + 47T^{2} \) |
| 53 | \( 1 - 0.690T + 53T^{2} \) |
| 59 | \( 1 + 9.44T + 59T^{2} \) |
| 61 | \( 1 - 11.2T + 61T^{2} \) |
| 67 | \( 1 + 12.1T + 67T^{2} \) |
| 71 | \( 1 + 7.39T + 71T^{2} \) |
| 73 | \( 1 - 0.601T + 73T^{2} \) |
| 79 | \( 1 + 10.3T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 7.29T + 89T^{2} \) |
| 97 | \( 1 + 13.8T + 97T^{2} \) |
| 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
−7.49701174640406893835768864184, −7.02751403477568059332019076890, −5.90223166122621583593044151779, −5.55959729945913423236149404185, −4.73870184590704379025168555073, −3.52199322106491034408788029813, −2.98893823744147074529520812322, −2.34875642004092433803056987412, −1.68395328218789227082989817477, 0,
1.68395328218789227082989817477, 2.34875642004092433803056987412, 2.98893823744147074529520812322, 3.52199322106491034408788029813, 4.73870184590704379025168555073, 5.55959729945913423236149404185, 5.90223166122621583593044151779, 7.02751403477568059332019076890, 7.49701174640406893835768864184
