| L(s) = 1 | + 1.44·3-s − 1.44·5-s + 0.445·7-s − 0.911·9-s − 2.33·11-s − 2.08·15-s + 6.89·17-s − 6.74·19-s + 0.643·21-s + 5.71·23-s − 2.91·25-s − 5.65·27-s − 8.14·29-s + 4.54·31-s − 3.37·33-s − 0.643·35-s − 8.07·37-s + 1.71·41-s + 3.86·43-s + 1.31·45-s + 0.829·47-s − 6.80·49-s + 9.97·51-s − 7.02·53-s + 3.37·55-s − 9.74·57-s + 3.59·59-s + ⋯ |
| L(s) = 1 | + 0.834·3-s − 0.646·5-s + 0.168·7-s − 0.303·9-s − 0.704·11-s − 0.539·15-s + 1.67·17-s − 1.54·19-s + 0.140·21-s + 1.19·23-s − 0.582·25-s − 1.08·27-s − 1.51·29-s + 0.815·31-s − 0.587·33-s − 0.108·35-s − 1.32·37-s + 0.267·41-s + 0.589·43-s + 0.196·45-s + 0.120·47-s − 0.971·49-s + 1.39·51-s − 0.965·53-s + 0.454·55-s − 1.29·57-s + 0.467·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 - 1.44T + 3T^{2} \) |
| 5 | \( 1 + 1.44T + 5T^{2} \) |
| 7 | \( 1 - 0.445T + 7T^{2} \) |
| 11 | \( 1 + 2.33T + 11T^{2} \) |
| 17 | \( 1 - 6.89T + 17T^{2} \) |
| 19 | \( 1 + 6.74T + 19T^{2} \) |
| 23 | \( 1 - 5.71T + 23T^{2} \) |
| 29 | \( 1 + 8.14T + 29T^{2} \) |
| 31 | \( 1 - 4.54T + 31T^{2} \) |
| 37 | \( 1 + 8.07T + 37T^{2} \) |
| 41 | \( 1 - 1.71T + 41T^{2} \) |
| 43 | \( 1 - 3.86T + 43T^{2} \) |
| 47 | \( 1 - 0.829T + 47T^{2} \) |
| 53 | \( 1 + 7.02T + 53T^{2} \) |
| 59 | \( 1 - 3.59T + 59T^{2} \) |
| 61 | \( 1 + 1.65T + 61T^{2} \) |
| 67 | \( 1 + 13.3T + 67T^{2} \) |
| 71 | \( 1 + 14.4T + 71T^{2} \) |
| 73 | \( 1 + 6.35T + 73T^{2} \) |
| 79 | \( 1 - 6.47T + 79T^{2} \) |
| 83 | \( 1 + 2.41T + 83T^{2} \) |
| 89 | \( 1 + 13.2T + 89T^{2} \) |
| 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
−8.418968400040825269160664786994, −7.75676872259688994074386363798, −7.31148742873779972355535284227, −6.07772790592961745437281393328, −5.34606336701190402587675282769, −4.34869624424099025713519019355, −3.44940667839515811880471296497, −2.82036653028143799702074733526, −1.68813178940736720674017509173, 0,
1.68813178940736720674017509173, 2.82036653028143799702074733526, 3.44940667839515811880471296497, 4.34869624424099025713519019355, 5.34606336701190402587675282769, 6.07772790592961745437281393328, 7.31148742873779972355535284227, 7.75676872259688994074386363798, 8.418968400040825269160664786994
