| L(s) = 1 | + 2-s + 3.43·3-s + 4-s + 2.13·5-s + 3.43·6-s − 0.845·7-s + 8-s + 8.77·9-s + 2.13·10-s − 4.69·11-s + 3.43·12-s + 6.42·13-s − 0.845·14-s + 7.32·15-s + 16-s − 5.11·17-s + 8.77·18-s − 1.16·19-s + 2.13·20-s − 2.90·21-s − 4.69·22-s + 1.86·23-s + 3.43·24-s − 0.449·25-s + 6.42·26-s + 19.8·27-s − 0.845·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.98·3-s + 0.5·4-s + 0.954·5-s + 1.40·6-s − 0.319·7-s + 0.353·8-s + 2.92·9-s + 0.674·10-s − 1.41·11-s + 0.990·12-s + 1.78·13-s − 0.225·14-s + 1.89·15-s + 0.250·16-s − 1.24·17-s + 2.06·18-s − 0.266·19-s + 0.477·20-s − 0.632·21-s − 1.00·22-s + 0.387·23-s + 0.700·24-s − 0.0898·25-s + 1.25·26-s + 3.81·27-s − 0.159·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6014 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.027918244\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.027918244\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 31 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| good | 3 | \( 1 - 3.43T + 3T^{2} \) |
| 5 | \( 1 - 2.13T + 5T^{2} \) |
| 7 | \( 1 + 0.845T + 7T^{2} \) |
| 11 | \( 1 + 4.69T + 11T^{2} \) |
| 13 | \( 1 - 6.42T + 13T^{2} \) |
| 17 | \( 1 + 5.11T + 17T^{2} \) |
| 19 | \( 1 + 1.16T + 19T^{2} \) |
| 23 | \( 1 - 1.86T + 23T^{2} \) |
| 29 | \( 1 + 4.03T + 29T^{2} \) |
| 37 | \( 1 + 3.99T + 37T^{2} \) |
| 41 | \( 1 - 6.39T + 41T^{2} \) |
| 43 | \( 1 - 10.6T + 43T^{2} \) |
| 47 | \( 1 - 1.11T + 47T^{2} \) |
| 53 | \( 1 - 0.569T + 53T^{2} \) |
| 59 | \( 1 - 3.02T + 59T^{2} \) |
| 61 | \( 1 + 4.84T + 61T^{2} \) |
| 67 | \( 1 + 13.5T + 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 - 11.6T + 73T^{2} \) |
| 79 | \( 1 + 7.85T + 79T^{2} \) |
| 83 | \( 1 - 12.9T + 83T^{2} \) |
| 89 | \( 1 + 9.38T + 89T^{2} \) |
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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
−8.054214669502529458364033396749, −7.51778705902544217062212749155, −6.62408124053114181867559520624, −6.03806533330269086195795501363, −5.07882860094269045474625784392, −4.15134195600275642798093518945, −3.58655429500695177385485378347, −2.68611057734529791835626023479, −2.27484691625306932797006371512, −1.40276451592676405933694322436,
1.40276451592676405933694322436, 2.27484691625306932797006371512, 2.68611057734529791835626023479, 3.58655429500695177385485378347, 4.15134195600275642798093518945, 5.07882860094269045474625784392, 6.03806533330269086195795501363, 6.62408124053114181867559520624, 7.51778705902544217062212749155, 8.054214669502529458364033396749
