| L(s) = 1 | + 2-s + 4-s − 5-s + 7-s + 8-s − 10-s − 11-s − 1.46·13-s + 14-s + 16-s − 3.46·17-s − 2.73·19-s − 20-s − 22-s − 1.26·23-s + 25-s − 1.46·26-s + 28-s + 4.73·29-s + 8.92·31-s + 32-s − 3.46·34-s − 35-s + 3.26·37-s − 2.73·38-s − 40-s + 4.73·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 0.447·5-s + 0.377·7-s + 0.353·8-s − 0.316·10-s − 0.301·11-s − 0.406·13-s + 0.267·14-s + 0.250·16-s − 0.840·17-s − 0.626·19-s − 0.223·20-s − 0.213·22-s − 0.264·23-s + 0.200·25-s − 0.287·26-s + 0.188·28-s + 0.878·29-s + 1.60·31-s + 0.176·32-s − 0.594·34-s − 0.169·35-s + 0.537·37-s − 0.443·38-s − 0.158·40-s + 0.739·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6930 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.800552655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.800552655\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + 1.46T + 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 2.73T + 19T^{2} \) |
| 23 | \( 1 + 1.26T + 23T^{2} \) |
| 29 | \( 1 - 4.73T + 29T^{2} \) |
| 31 | \( 1 - 8.92T + 31T^{2} \) |
| 37 | \( 1 - 3.26T + 37T^{2} \) |
| 41 | \( 1 - 4.73T + 41T^{2} \) |
| 43 | \( 1 + 4.92T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 4.73T + 53T^{2} \) |
| 59 | \( 1 - 13.8T + 59T^{2} \) |
| 61 | \( 1 - 2T + 61T^{2} \) |
| 67 | \( 1 + 10.9T + 67T^{2} \) |
| 71 | \( 1 + 9.46T + 71T^{2} \) |
| 73 | \( 1 - 11.4T + 73T^{2} \) |
| 79 | \( 1 - 6.73T + 79T^{2} \) |
| 83 | \( 1 - 4.39T + 83T^{2} \) |
| 89 | \( 1 - 15.4T + 89T^{2} \) |
| 97 | \( 1 - 5.80T + 97T^{2} \) |
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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
−7.978964998614973655197450280283, −7.14013886386978090854371717682, −6.52223053915333230002798580287, −5.85517046709022550412858726142, −4.82683838986969613838725365196, −4.54352712636618204622527396983, −3.70595806652354631805188201060, −2.71638556143364600168599886768, −2.10655789273202128722845380458, −0.75319626833221166755646784724,
0.75319626833221166755646784724, 2.10655789273202128722845380458, 2.71638556143364600168599886768, 3.70595806652354631805188201060, 4.54352712636618204622527396983, 4.82683838986969613838725365196, 5.85517046709022550412858726142, 6.52223053915333230002798580287, 7.14013886386978090854371717682, 7.978964998614973655197450280283
