| L(s) = 1 | + 2-s + 4-s − 3.70·5-s + 7-s + 8-s − 3.70·10-s + 4·11-s + 5.70·13-s + 14-s + 16-s − 4·17-s − 5.40·19-s − 3.70·20-s + 4·22-s − 23-s + 8.70·25-s + 5.70·26-s + 28-s − 0.298·29-s − 2·31-s + 32-s − 4·34-s − 3.70·35-s + 4.29·37-s − 5.40·38-s − 3.70·40-s + 0.298·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.65·5-s + 0.377·7-s + 0.353·8-s − 1.17·10-s + 1.20·11-s + 1.58·13-s + 0.267·14-s + 0.250·16-s − 0.970·17-s − 1.23·19-s − 0.827·20-s + 0.852·22-s − 0.208·23-s + 1.74·25-s + 1.11·26-s + 0.188·28-s − 0.0554·29-s − 0.359·31-s + 0.176·32-s − 0.685·34-s − 0.625·35-s + 0.706·37-s − 0.876·38-s − 0.585·40-s + 0.0466·41-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
| \(L(1)\) |
\(\approx\) |
\(2.406239016\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.406239016\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + 3.70T + 5T^{2} \) |
| 11 | \( 1 - 4T + 11T^{2} \) |
| 13 | \( 1 - 5.70T + 13T^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + 5.40T + 19T^{2} \) |
| 29 | \( 1 + 0.298T + 29T^{2} \) |
| 31 | \( 1 + 2T + 31T^{2} \) |
| 37 | \( 1 - 4.29T + 37T^{2} \) |
| 41 | \( 1 - 0.298T + 41T^{2} \) |
| 43 | \( 1 + 1.70T + 43T^{2} \) |
| 47 | \( 1 - 11.1T + 47T^{2} \) |
| 53 | \( 1 - 9.40T + 53T^{2} \) |
| 59 | \( 1 - 7.40T + 59T^{2} \) |
| 61 | \( 1 + 2T + 61T^{2} \) |
| 67 | \( 1 - 14.8T + 67T^{2} \) |
| 71 | \( 1 - 7.40T + 71T^{2} \) |
| 73 | \( 1 - 1.40T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + 13.4T + 83T^{2} \) |
| 89 | \( 1 - 11.4T + 89T^{2} \) |
| 97 | \( 1 - 6.29T + 97T^{2} \) |
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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
−8.500622646010457550698242244657, −8.146652912397608881474640982753, −7.05225369316589681007727539576, −6.59469215707706262623068516039, −5.71794245741845818685227151327, −4.48266346907206762502895641965, −4.00730420632544595064044240466, −3.59550586560046086395018239392, −2.20284918099536179505614135847, −0.882849108044602374910707935437,
0.882849108044602374910707935437, 2.20284918099536179505614135847, 3.59550586560046086395018239392, 4.00730420632544595064044240466, 4.48266346907206762502895641965, 5.71794245741845818685227151327, 6.59469215707706262623068516039, 7.05225369316589681007727539576, 8.146652912397608881474640982753, 8.500622646010457550698242244657
