| L(s) = 1 | + 2.44·3-s + 5-s − 2·7-s + 2.99·9-s + 4.44·11-s − 13-s + 2.44·15-s + 6.89·17-s − 0.449·19-s − 4.89·21-s − 6.44·23-s + 25-s + 4·29-s + 4.44·31-s + 10.8·33-s − 2·35-s + 4.89·37-s − 2.44·39-s + 10.8·41-s − 11.3·43-s + 2.99·45-s + 2·47-s − 3·49-s + 16.8·51-s + 1.10·53-s + 4.44·55-s − 1.10·57-s + ⋯ |
| L(s) = 1 | + 1.41·3-s + 0.447·5-s − 0.755·7-s + 0.999·9-s + 1.34·11-s − 0.277·13-s + 0.632·15-s + 1.67·17-s − 0.103·19-s − 1.06·21-s − 1.34·23-s + 0.200·25-s + 0.742·29-s + 0.799·31-s + 1.89·33-s − 0.338·35-s + 0.805·37-s − 0.392·39-s + 1.70·41-s − 1.73·43-s + 0.447·45-s + 0.291·47-s − 0.428·49-s + 2.36·51-s + 0.151·53-s + 0.599·55-s − 0.145·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.794436604\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.794436604\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 - 2.44T + 3T^{2} \) |
| 7 | \( 1 + 2T + 7T^{2} \) |
| 11 | \( 1 - 4.44T + 11T^{2} \) |
| 17 | \( 1 - 6.89T + 17T^{2} \) |
| 19 | \( 1 + 0.449T + 19T^{2} \) |
| 23 | \( 1 + 6.44T + 23T^{2} \) |
| 29 | \( 1 - 4T + 29T^{2} \) |
| 31 | \( 1 - 4.44T + 31T^{2} \) |
| 37 | \( 1 - 4.89T + 37T^{2} \) |
| 41 | \( 1 - 10.8T + 41T^{2} \) |
| 43 | \( 1 + 11.3T + 43T^{2} \) |
| 47 | \( 1 - 2T + 47T^{2} \) |
| 53 | \( 1 - 1.10T + 53T^{2} \) |
| 59 | \( 1 + 9.34T + 59T^{2} \) |
| 61 | \( 1 + 5.79T + 61T^{2} \) |
| 67 | \( 1 - 5.10T + 67T^{2} \) |
| 71 | \( 1 - 3.55T + 71T^{2} \) |
| 73 | \( 1 + 14.6T + 73T^{2} \) |
| 79 | \( 1 + 4.89T + 79T^{2} \) |
| 83 | \( 1 + 2T + 83T^{2} \) |
| 89 | \( 1 - 6T + 89T^{2} \) |
| 97 | \( 1 - 7.79T + 97T^{2} \) |
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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
−9.694865661692728389419036769782, −9.256412962290186198264348315753, −8.294543470082443606721732593525, −7.64600853140530620705136686083, −6.56044804036594198344249391208, −5.84928530986272105417580384848, −4.34801535491750197447662144844, −3.46116090705471850123679945848, −2.68803273995179098117398992176, −1.42054061096595350834882889504,
1.42054061096595350834882889504, 2.68803273995179098117398992176, 3.46116090705471850123679945848, 4.34801535491750197447662144844, 5.84928530986272105417580384848, 6.56044804036594198344249391208, 7.64600853140530620705136686083, 8.294543470082443606721732593525, 9.256412962290186198264348315753, 9.694865661692728389419036769782
