| L(s) = 1 | − 2-s + 4-s + 2·5-s + 7-s − 8-s − 2·10-s + 4·11-s − 2·13-s − 14-s + 16-s + 6·17-s + 4·19-s + 2·20-s − 4·22-s + 23-s − 25-s + 2·26-s + 28-s + 2·29-s − 8·31-s − 32-s − 6·34-s + 2·35-s + 6·37-s − 4·38-s − 2·40-s + 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.377·7-s − 0.353·8-s − 0.632·10-s + 1.20·11-s − 0.554·13-s − 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.917·19-s + 0.447·20-s − 0.852·22-s + 0.208·23-s − 1/5·25-s + 0.392·26-s + 0.188·28-s + 0.371·29-s − 1.43·31-s − 0.176·32-s − 1.02·34-s + 0.338·35-s + 0.986·37-s − 0.648·38-s − 0.316·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2898 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.911394957\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.911394957\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.129534828526019448146707365409, −7.87331920702619621518265165883, −7.48778741343048056178566317467, −6.51047619711733273708478483465, −5.79997409848774960709448384956, −5.11412604771931773767220495242, −3.90651977124452563863091351072, −2.91289799284385187077537566699, −1.80993947891562273249411861323, −1.03160171757055072259869718966,
1.03160171757055072259869718966, 1.80993947891562273249411861323, 2.91289799284385187077537566699, 3.90651977124452563863091351072, 5.11412604771931773767220495242, 5.79997409848774960709448384956, 6.51047619711733273708478483465, 7.48778741343048056178566317467, 7.87331920702619621518265165883, 9.129534828526019448146707365409
