L(s) = 1 | + 3.41·3-s − 0.828·7-s + 8.65·9-s − 2·11-s + 6.24·13-s − 0.828·17-s − 19-s − 2.82·21-s + 6·23-s + 19.3·27-s − 6.48·29-s − 6.82·31-s − 6.82·33-s + 1.75·37-s + 21.3·39-s + 3.65·41-s − 4.82·43-s + 4.82·47-s − 6.31·49-s − 2.82·51-s − 9.07·53-s − 3.41·57-s + 13.6·59-s − 13.6·61-s − 7.17·63-s + 3.41·67-s + 20.4·69-s + ⋯ |
L(s) = 1 | + 1.97·3-s − 0.313·7-s + 2.88·9-s − 0.603·11-s + 1.73·13-s − 0.200·17-s − 0.229·19-s − 0.617·21-s + 1.25·23-s + 3.71·27-s − 1.20·29-s − 1.22·31-s − 1.18·33-s + 0.288·37-s + 3.41·39-s + 0.571·41-s − 0.736·43-s + 0.704·47-s − 0.901·49-s − 0.396·51-s − 1.24·53-s − 0.452·57-s + 1.77·59-s − 1.74·61-s − 0.903·63-s + 0.417·67-s + 2.46·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.729598454\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.729598454\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 3.41T + 3T^{2} \) |
| 7 | \( 1 + 0.828T + 7T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 - 6.24T + 13T^{2} \) |
| 17 | \( 1 + 0.828T + 17T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + 6.48T + 29T^{2} \) |
| 31 | \( 1 + 6.82T + 31T^{2} \) |
| 37 | \( 1 - 1.75T + 37T^{2} \) |
| 41 | \( 1 - 3.65T + 41T^{2} \) |
| 43 | \( 1 + 4.82T + 43T^{2} \) |
| 47 | \( 1 - 4.82T + 47T^{2} \) |
| 53 | \( 1 + 9.07T + 53T^{2} \) |
| 59 | \( 1 - 13.6T + 59T^{2} \) |
| 61 | \( 1 + 13.6T + 61T^{2} \) |
| 67 | \( 1 - 3.41T + 67T^{2} \) |
| 71 | \( 1 - 5.17T + 71T^{2} \) |
| 73 | \( 1 - 2.48T + 73T^{2} \) |
| 79 | \( 1 - 1.65T + 79T^{2} \) |
| 83 | \( 1 - 13.3T + 83T^{2} \) |
| 89 | \( 1 + 6.48T + 89T^{2} \) |
| 97 | \( 1 - 10.2T + 97T^{2} \) |
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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
−9.105350401243179652686633724661, −8.518153926659759368142620055655, −7.82274375813984079578170040733, −7.12570631103319291875353444803, −6.22801289309547539997526798333, −4.95184551981428610559216145997, −3.80120206116264116607526388645, −3.39063017175449050963740697219, −2.38876468739221003337205171349, −1.38357515986915209515578547703,
1.38357515986915209515578547703, 2.38876468739221003337205171349, 3.39063017175449050963740697219, 3.80120206116264116607526388645, 4.95184551981428610559216145997, 6.22801289309547539997526798333, 7.12570631103319291875353444803, 7.82274375813984079578170040733, 8.518153926659759368142620055655, 9.105350401243179652686633724661