| L(s) = 1 | + 2-s − 1.30·3-s + 4-s + 2.66·5-s − 1.30·6-s − 4.20·7-s + 8-s − 1.30·9-s + 2.66·10-s − 2.85·11-s − 1.30·12-s + 5.25·13-s − 4.20·14-s − 3.47·15-s + 16-s − 3.66·17-s − 1.30·18-s − 2.56·19-s + 2.66·20-s + 5.46·21-s − 2.85·22-s + 23-s − 1.30·24-s + 2.11·25-s + 5.25·26-s + 5.60·27-s − 4.20·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.750·3-s + 0.5·4-s + 1.19·5-s − 0.530·6-s − 1.58·7-s + 0.353·8-s − 0.436·9-s + 0.843·10-s − 0.860·11-s − 0.375·12-s + 1.45·13-s − 1.12·14-s − 0.896·15-s + 0.250·16-s − 0.888·17-s − 0.308·18-s − 0.589·19-s + 0.596·20-s + 1.19·21-s − 0.608·22-s + 0.208·23-s − 0.265·24-s + 0.423·25-s + 1.03·26-s + 1.07·27-s − 0.794·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.047494631\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.047494631\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 - T \) |
| good | 3 | \( 1 + 1.30T + 3T^{2} \) |
| 5 | \( 1 - 2.66T + 5T^{2} \) |
| 7 | \( 1 + 4.20T + 7T^{2} \) |
| 11 | \( 1 + 2.85T + 11T^{2} \) |
| 13 | \( 1 - 5.25T + 13T^{2} \) |
| 17 | \( 1 + 3.66T + 17T^{2} \) |
| 19 | \( 1 + 2.56T + 19T^{2} \) |
| 29 | \( 1 - 1.86T + 29T^{2} \) |
| 31 | \( 1 - 9.92T + 31T^{2} \) |
| 37 | \( 1 - 0.702T + 37T^{2} \) |
| 41 | \( 1 + 1.57T + 41T^{2} \) |
| 43 | \( 1 + 3.65T + 43T^{2} \) |
| 47 | \( 1 + 9.81T + 47T^{2} \) |
| 53 | \( 1 + 1.81T + 53T^{2} \) |
| 59 | \( 1 - 7.25T + 59T^{2} \) |
| 61 | \( 1 - 12.6T + 61T^{2} \) |
| 67 | \( 1 + 15.7T + 67T^{2} \) |
| 71 | \( 1 - 0.378T + 71T^{2} \) |
| 73 | \( 1 - 11.9T + 73T^{2} \) |
| 79 | \( 1 - 12.9T + 79T^{2} \) |
| 83 | \( 1 - 8.73T + 83T^{2} \) |
| 89 | \( 1 + 4.30T + 89T^{2} \) |
| 97 | \( 1 - 17.8T + 97T^{2} \) |
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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
−8.148893330513265626716918044198, −6.74255641976981151451141009331, −6.38869232373490926924028163194, −6.10906159586339347290074694551, −5.36254718959064187361488277147, −4.64833340865715891669367244962, −3.53884991708912633349834999578, −2.86363564660288996709392949023, −2.08446467253850709890626555619, −0.67262395589064802636501342183,
0.67262395589064802636501342183, 2.08446467253850709890626555619, 2.86363564660288996709392949023, 3.53884991708912633349834999578, 4.64833340865715891669367244962, 5.36254718959064187361488277147, 6.10906159586339347290074694551, 6.38869232373490926924028163194, 6.74255641976981151451141009331, 8.148893330513265626716918044198
