| L(s) = 1 | + 0.554·3-s + 1.10·7-s − 2.69·9-s + 2.71·11-s − 1.69·13-s + 1.10·17-s − 4.98·19-s + 0.615·21-s − 23-s − 3.15·27-s + 6.34·29-s + 5.13·31-s + 1.50·33-s + 5.70·37-s − 0.939·39-s + 1.26·41-s − 1.90·43-s + 4.08·47-s − 5.76·49-s + 0.615·51-s + 3.78·53-s − 2.76·57-s + 5.59·59-s + 8.37·61-s − 2.98·63-s + 10.7·67-s − 0.554·69-s + ⋯ |
| L(s) = 1 | + 0.320·3-s + 0.419·7-s − 0.897·9-s + 0.818·11-s − 0.469·13-s + 0.269·17-s − 1.14·19-s + 0.134·21-s − 0.208·23-s − 0.607·27-s + 1.17·29-s + 0.922·31-s + 0.262·33-s + 0.937·37-s − 0.150·39-s + 0.198·41-s − 0.290·43-s + 0.596·47-s − 0.824·49-s + 0.0862·51-s + 0.519·53-s − 0.366·57-s + 0.727·59-s + 1.07·61-s − 0.376·63-s + 1.31·67-s − 0.0668·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.088966735\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.088966735\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - 0.554T + 3T^{2} \) |
| 7 | \( 1 - 1.10T + 7T^{2} \) |
| 11 | \( 1 - 2.71T + 11T^{2} \) |
| 13 | \( 1 + 1.69T + 13T^{2} \) |
| 17 | \( 1 - 1.10T + 17T^{2} \) |
| 19 | \( 1 + 4.98T + 19T^{2} \) |
| 29 | \( 1 - 6.34T + 29T^{2} \) |
| 31 | \( 1 - 5.13T + 31T^{2} \) |
| 37 | \( 1 - 5.70T + 37T^{2} \) |
| 41 | \( 1 - 1.26T + 41T^{2} \) |
| 43 | \( 1 + 1.90T + 43T^{2} \) |
| 47 | \( 1 - 4.08T + 47T^{2} \) |
| 53 | \( 1 - 3.78T + 53T^{2} \) |
| 59 | \( 1 - 5.59T + 59T^{2} \) |
| 61 | \( 1 - 8.37T + 61T^{2} \) |
| 67 | \( 1 - 10.7T + 67T^{2} \) |
| 71 | \( 1 + 12.1T + 71T^{2} \) |
| 73 | \( 1 - 15.3T + 73T^{2} \) |
| 79 | \( 1 - 11.0T + 79T^{2} \) |
| 83 | \( 1 + 7.64T + 83T^{2} \) |
| 89 | \( 1 + 4.17T + 89T^{2} \) |
| 97 | \( 1 - 9.50T + 97T^{2} \) |
| 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
−8.345035428800039798716736947568, −7.80970254614493624348146866642, −6.75282584271519609260758219095, −6.25373295417899084681489170106, −5.35996569166983094883843315367, −4.53878939064355490826793122550, −3.80676244996758125839049170237, −2.79097164746481690916482024466, −2.08184726446362094521438553331, −0.78859558304663548036663875697,
0.78859558304663548036663875697, 2.08184726446362094521438553331, 2.79097164746481690916482024466, 3.80676244996758125839049170237, 4.53878939064355490826793122550, 5.35996569166983094883843315367, 6.25373295417899084681489170106, 6.75282584271519609260758219095, 7.80970254614493624348146866642, 8.345035428800039798716736947568
