| L(s) = 1 | − 2-s + 4-s + 3.79·5-s − 8-s − 3.79·10-s − 11-s − 0.361·13-s + 16-s − 4.11·17-s − 4.15·19-s + 3.79·20-s + 22-s − 0.542·23-s + 9.42·25-s + 0.361·26-s − 0.767·29-s + 8.80·31-s − 32-s + 4.11·34-s + 2.28·37-s + 4.15·38-s − 3.79·40-s + 9.13·41-s − 10.1·43-s − 44-s + 0.542·46-s − 2.98·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 1.69·5-s − 0.353·8-s − 1.20·10-s − 0.301·11-s − 0.100·13-s + 0.250·16-s − 0.998·17-s − 0.954·19-s + 0.849·20-s + 0.213·22-s − 0.113·23-s + 1.88·25-s + 0.0707·26-s − 0.142·29-s + 1.58·31-s − 0.176·32-s + 0.705·34-s + 0.375·37-s + 0.674·38-s − 0.600·40-s + 1.42·41-s − 1.55·43-s − 0.150·44-s + 0.0800·46-s − 0.435·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9702 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.931876099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.931876099\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 3.79T + 5T^{2} \) |
| 13 | \( 1 + 0.361T + 13T^{2} \) |
| 17 | \( 1 + 4.11T + 17T^{2} \) |
| 19 | \( 1 + 4.15T + 19T^{2} \) |
| 23 | \( 1 + 0.542T + 23T^{2} \) |
| 29 | \( 1 + 0.767T + 29T^{2} \) |
| 31 | \( 1 - 8.80T + 31T^{2} \) |
| 37 | \( 1 - 2.28T + 37T^{2} \) |
| 41 | \( 1 - 9.13T + 41T^{2} \) |
| 43 | \( 1 + 10.1T + 43T^{2} \) |
| 47 | \( 1 + 2.98T + 47T^{2} \) |
| 53 | \( 1 - 12.4T + 53T^{2} \) |
| 59 | \( 1 - 2.25T + 59T^{2} \) |
| 61 | \( 1 + 12.2T + 61T^{2} \) |
| 67 | \( 1 - 0.603T + 67T^{2} \) |
| 71 | \( 1 - 6.87T + 71T^{2} \) |
| 73 | \( 1 + 9.45T + 73T^{2} \) |
| 79 | \( 1 - 0.0321T + 79T^{2} \) |
| 83 | \( 1 - 10.8T + 83T^{2} \) |
| 89 | \( 1 + 1.29T + 89T^{2} \) |
| 97 | \( 1 - 18.2T + 97T^{2} \) |
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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
−7.74770932481054741518432385320, −6.87126010716069601600700689012, −6.30913808439706090038234111524, −5.92378466259394596813503631841, −5.01222593809135285742669905739, −4.35130919691371490962084332927, −3.05863509887566335105862612028, −2.29729779099725732746579763261, −1.85908324218875776374963807520, −0.72494741398765867292009806949,
0.72494741398765867292009806949, 1.85908324218875776374963807520, 2.29729779099725732746579763261, 3.05863509887566335105862612028, 4.35130919691371490962084332927, 5.01222593809135285742669905739, 5.92378466259394596813503631841, 6.30913808439706090038234111524, 6.87126010716069601600700689012, 7.74770932481054741518432385320
