| L(s) = 1 | + 2.73·3-s + 2·7-s + 4.46·9-s + 3.46·11-s + 2.73·13-s + 3.46·17-s − 19-s + 5.46·21-s − 3.46·23-s + 3.99·27-s + 3.46·29-s + 1.46·31-s + 9.46·33-s − 6.73·37-s + 7.46·39-s − 6·41-s − 4.92·43-s + 12.9·47-s − 3·49-s + 9.46·51-s + 10.7·53-s − 2.73·57-s − 6.92·59-s + 12.3·61-s + 8.92·63-s + 6.73·67-s − 9.46·69-s + ⋯ |
| L(s) = 1 | + 1.57·3-s + 0.755·7-s + 1.48·9-s + 1.04·11-s + 0.757·13-s + 0.840·17-s − 0.229·19-s + 1.19·21-s − 0.722·23-s + 0.769·27-s + 0.643·29-s + 0.262·31-s + 1.64·33-s − 1.10·37-s + 1.19·39-s − 0.937·41-s − 0.751·43-s + 1.88·47-s − 0.428·49-s + 1.32·51-s + 1.47·53-s − 0.361·57-s − 0.901·59-s + 1.58·61-s + 1.12·63-s + 0.822·67-s − 1.13·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.936337232\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.936337232\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 + T \) |
| good | 3 | \( 1 - 2.73T + 3T^{2} \) |
| 7 | \( 1 - 2T + 7T^{2} \) |
| 11 | \( 1 - 3.46T + 11T^{2} \) |
| 13 | \( 1 - 2.73T + 13T^{2} \) |
| 17 | \( 1 - 3.46T + 17T^{2} \) |
| 23 | \( 1 + 3.46T + 23T^{2} \) |
| 29 | \( 1 - 3.46T + 29T^{2} \) |
| 31 | \( 1 - 1.46T + 31T^{2} \) |
| 37 | \( 1 + 6.73T + 37T^{2} \) |
| 41 | \( 1 + 6T + 41T^{2} \) |
| 43 | \( 1 + 4.92T + 43T^{2} \) |
| 47 | \( 1 - 12.9T + 47T^{2} \) |
| 53 | \( 1 - 10.7T + 53T^{2} \) |
| 59 | \( 1 + 6.92T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 6.73T + 67T^{2} \) |
| 71 | \( 1 - 2.53T + 71T^{2} \) |
| 73 | \( 1 - 0.535T + 73T^{2} \) |
| 79 | \( 1 + 2.92T + 79T^{2} \) |
| 83 | \( 1 - 3.46T + 83T^{2} \) |
| 89 | \( 1 + 15.4T + 89T^{2} \) |
| 97 | \( 1 - 16.5T + 97T^{2} \) |
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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
−8.184117279333769463752723412714, −7.31215968811451396352487736798, −6.71348595204637031534107314995, −5.80738118101595791861275433255, −4.91952098690499083583962902888, −3.89142157533166258170583568292, −3.70798630141481108906072967617, −2.68785692148324296686400873187, −1.83826380802147783647719534327, −1.15132721047124141831873828615,
1.15132721047124141831873828615, 1.83826380802147783647719534327, 2.68785692148324296686400873187, 3.70798630141481108906072967617, 3.89142157533166258170583568292, 4.91952098690499083583962902888, 5.80738118101595791861275433255, 6.71348595204637031534107314995, 7.31215968811451396352487736798, 8.184117279333769463752723412714
