| L(s) = 1 | + 3.12·3-s − 2.38·5-s − 0.375·7-s + 6.73·9-s + 5.91·11-s + 1.56·13-s − 7.44·15-s + 5.73·17-s + 3.33·19-s − 1.17·21-s − 2.78·23-s + 0.691·25-s + 11.6·27-s + 6.29·29-s − 6.31·31-s + 18.4·33-s + 0.896·35-s + 1.67·37-s + 4.87·39-s − 1.41·41-s + 2.00·43-s − 16.0·45-s − 5.32·47-s − 6.85·49-s + 17.9·51-s + 9.72·53-s − 14.1·55-s + ⋯ |
| L(s) = 1 | + 1.80·3-s − 1.06·5-s − 0.142·7-s + 2.24·9-s + 1.78·11-s + 0.432·13-s − 1.92·15-s + 1.39·17-s + 0.764·19-s − 0.256·21-s − 0.580·23-s + 0.138·25-s + 2.24·27-s + 1.16·29-s − 1.13·31-s + 3.21·33-s + 0.151·35-s + 0.274·37-s + 0.779·39-s − 0.220·41-s + 0.305·43-s − 2.39·45-s − 0.777·47-s − 0.979·49-s + 2.50·51-s + 1.33·53-s − 1.90·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.341315347\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.341315347\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 503 | \( 1 + T \) |
| good | 3 | \( 1 - 3.12T + 3T^{2} \) |
| 5 | \( 1 + 2.38T + 5T^{2} \) |
| 7 | \( 1 + 0.375T + 7T^{2} \) |
| 11 | \( 1 - 5.91T + 11T^{2} \) |
| 13 | \( 1 - 1.56T + 13T^{2} \) |
| 17 | \( 1 - 5.73T + 17T^{2} \) |
| 19 | \( 1 - 3.33T + 19T^{2} \) |
| 23 | \( 1 + 2.78T + 23T^{2} \) |
| 29 | \( 1 - 6.29T + 29T^{2} \) |
| 31 | \( 1 + 6.31T + 31T^{2} \) |
| 37 | \( 1 - 1.67T + 37T^{2} \) |
| 41 | \( 1 + 1.41T + 41T^{2} \) |
| 43 | \( 1 - 2.00T + 43T^{2} \) |
| 47 | \( 1 + 5.32T + 47T^{2} \) |
| 53 | \( 1 - 9.72T + 53T^{2} \) |
| 59 | \( 1 + 7.28T + 59T^{2} \) |
| 61 | \( 1 + 1.71T + 61T^{2} \) |
| 67 | \( 1 - 14.3T + 67T^{2} \) |
| 71 | \( 1 + 4.72T + 71T^{2} \) |
| 73 | \( 1 + 15.2T + 73T^{2} \) |
| 79 | \( 1 - 6.17T + 79T^{2} \) |
| 83 | \( 1 + 4.91T + 83T^{2} \) |
| 89 | \( 1 + 9.88T + 89T^{2} \) |
| 97 | \( 1 - 6.38T + 97T^{2} \) |
| 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
−7.86973645326271969692839919578, −7.39624308002993429900505785948, −6.75830379209020758449031986735, −5.86660610461194306469864495459, −4.63270767748539969143857272083, −3.90086671480348136888527545537, −3.53903313687061157531669846787, −2.99350537581937421775006196762, −1.76364473063336181090582127936, −1.03906706613242362058211482073,
1.03906706613242362058211482073, 1.76364473063336181090582127936, 2.99350537581937421775006196762, 3.53903313687061157531669846787, 3.90086671480348136888527545537, 4.63270767748539969143857272083, 5.86660610461194306469864495459, 6.75830379209020758449031986735, 7.39624308002993429900505785948, 7.86973645326271969692839919578
