| L(s) = 1 | − 3.04·3-s + 5-s − 7-s + 6.24·9-s − 11-s + 4.52·13-s − 3.04·15-s + 2.52·17-s + 7.36·19-s + 3.04·21-s + 0.320·23-s + 25-s − 9.88·27-s + 5.56·29-s + 5.77·31-s + 3.04·33-s − 35-s − 4.81·37-s − 13.7·39-s + 6.00·41-s + 9.04·43-s + 6.24·45-s − 2.17·47-s + 49-s − 7.68·51-s + 1.67·53-s − 55-s + ⋯ |
| L(s) = 1 | − 1.75·3-s + 0.447·5-s − 0.377·7-s + 2.08·9-s − 0.301·11-s + 1.25·13-s − 0.785·15-s + 0.613·17-s + 1.69·19-s + 0.663·21-s + 0.0667·23-s + 0.200·25-s − 1.90·27-s + 1.03·29-s + 1.03·31-s + 0.529·33-s − 0.169·35-s − 0.792·37-s − 2.20·39-s + 0.938·41-s + 1.37·43-s + 0.931·45-s − 0.316·47-s + 0.142·49-s − 1.07·51-s + 0.230·53-s − 0.134·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.358619338\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.358619338\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 3 | \( 1 + 3.04T + 3T^{2} \) |
| 13 | \( 1 - 4.52T + 13T^{2} \) |
| 17 | \( 1 - 2.52T + 17T^{2} \) |
| 19 | \( 1 - 7.36T + 19T^{2} \) |
| 23 | \( 1 - 0.320T + 23T^{2} \) |
| 29 | \( 1 - 5.56T + 29T^{2} \) |
| 31 | \( 1 - 5.77T + 31T^{2} \) |
| 37 | \( 1 + 4.81T + 37T^{2} \) |
| 41 | \( 1 - 6.00T + 41T^{2} \) |
| 43 | \( 1 - 9.04T + 43T^{2} \) |
| 47 | \( 1 + 2.17T + 47T^{2} \) |
| 53 | \( 1 - 1.67T + 53T^{2} \) |
| 59 | \( 1 + 7.16T + 59T^{2} \) |
| 61 | \( 1 + 5.77T + 61T^{2} \) |
| 67 | \( 1 + 2.83T + 67T^{2} \) |
| 71 | \( 1 - 3.35T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 8.11T + 79T^{2} \) |
| 83 | \( 1 - 12.8T + 83T^{2} \) |
| 89 | \( 1 - 4.74T + 89T^{2} \) |
| 97 | \( 1 - 6.82T + 97T^{2} \) |
| 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
−7.81271610990205827595660441555, −7.20954820728435550767282930084, −6.22092505958973527865184566974, −6.11499001391103682341134031464, −5.29525798023197198361727010277, −4.74725096145014401517255877078, −3.74906224335799641952246258726, −2.83451778382137460366741460835, −1.34752569458907201215477247187, −0.77513884588939239289993798016,
0.77513884588939239289993798016, 1.34752569458907201215477247187, 2.83451778382137460366741460835, 3.74906224335799641952246258726, 4.74725096145014401517255877078, 5.29525798023197198361727010277, 6.11499001391103682341134031464, 6.22092505958973527865184566974, 7.20954820728435550767282930084, 7.81271610990205827595660441555
