| L(s) = 1 | + 2-s − 3-s + 4-s − 6-s + 8-s + 9-s − 6.24·11-s − 12-s + 0.171·13-s + 16-s − 4.65·17-s + 18-s + 2.58·19-s − 6.24·22-s + 1.58·23-s − 24-s + 0.171·26-s − 27-s − 10.6·29-s + 7.24·31-s + 32-s + 6.24·33-s − 4.65·34-s + 36-s + 0.242·37-s + 2.58·38-s − 0.171·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 0.5·4-s − 0.408·6-s + 0.353·8-s + 0.333·9-s − 1.88·11-s − 0.288·12-s + 0.0475·13-s + 0.250·16-s − 1.12·17-s + 0.235·18-s + 0.593·19-s − 1.33·22-s + 0.330·23-s − 0.204·24-s + 0.0336·26-s − 0.192·27-s − 1.97·29-s + 1.30·31-s + 0.176·32-s + 1.08·33-s − 0.798·34-s + 0.166·36-s + 0.0398·37-s + 0.419·38-s − 0.0274·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.909614749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.909614749\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 6.24T + 11T^{2} \) |
| 13 | \( 1 - 0.171T + 13T^{2} \) |
| 17 | \( 1 + 4.65T + 17T^{2} \) |
| 19 | \( 1 - 2.58T + 19T^{2} \) |
| 23 | \( 1 - 1.58T + 23T^{2} \) |
| 29 | \( 1 + 10.6T + 29T^{2} \) |
| 31 | \( 1 - 7.24T + 31T^{2} \) |
| 37 | \( 1 - 0.242T + 37T^{2} \) |
| 41 | \( 1 - 7T + 41T^{2} \) |
| 43 | \( 1 + 3.58T + 43T^{2} \) |
| 47 | \( 1 - 8T + 47T^{2} \) |
| 53 | \( 1 - 2.65T + 53T^{2} \) |
| 59 | \( 1 + 1.58T + 59T^{2} \) |
| 61 | \( 1 - 2.65T + 61T^{2} \) |
| 67 | \( 1 - 9.65T + 67T^{2} \) |
| 71 | \( 1 + 2T + 71T^{2} \) |
| 73 | \( 1 - 8.24T + 73T^{2} \) |
| 79 | \( 1 - 6.24T + 79T^{2} \) |
| 83 | \( 1 + 5.58T + 83T^{2} \) |
| 89 | \( 1 - 15.3T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−7.66263692450090686510326776552, −7.20631049385956534606659243840, −6.35749665209084826093271008217, −5.64424219206451047630738744027, −5.14767662828047935785018067411, −4.51227245967404926376717183911, −3.64586933829706876456116285702, −2.67878774145804997166649562950, −2.05887782358820700963961461289, −0.61575118696838810207196711915,
0.61575118696838810207196711915, 2.05887782358820700963961461289, 2.67878774145804997166649562950, 3.64586933829706876456116285702, 4.51227245967404926376717183911, 5.14767662828047935785018067411, 5.64424219206451047630738744027, 6.35749665209084826093271008217, 7.20631049385956534606659243840, 7.66263692450090686510326776552
