| L(s) = 1 | − 2.56·2-s − 1.43·4-s − 25.6·7-s + 24.1·8-s + 11·11-s − 8.38·13-s + 65.7·14-s − 50.4·16-s + 8.44·17-s + 19.5·19-s − 28.1·22-s + 189.·23-s + 21.4·26-s + 36.9·28-s − 235.·29-s + 46.4·31-s − 64.2·32-s − 21.6·34-s − 72.0·37-s − 50.0·38-s + 273.·41-s + 375.·43-s − 15.8·44-s − 485.·46-s − 266.·47-s + 316.·49-s + 12.0·52-s + ⋯ |
| L(s) = 1 | − 0.905·2-s − 0.179·4-s − 1.38·7-s + 1.06·8-s + 0.301·11-s − 0.178·13-s + 1.25·14-s − 0.787·16-s + 0.120·17-s + 0.236·19-s − 0.273·22-s + 1.71·23-s + 0.162·26-s + 0.249·28-s − 1.50·29-s + 0.268·31-s − 0.354·32-s − 0.109·34-s − 0.319·37-s − 0.213·38-s + 1.04·41-s + 1.33·43-s − 0.0542·44-s − 1.55·46-s − 0.828·47-s + 0.922·49-s + 0.0321·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.6610614141\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6610614141\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 2.56T + 8T^{2} \) |
| 7 | \( 1 + 25.6T + 343T^{2} \) |
| 13 | \( 1 + 8.38T + 2.19e3T^{2} \) |
| 17 | \( 1 - 8.44T + 4.91e3T^{2} \) |
| 19 | \( 1 - 19.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 189.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 235.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 46.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 72.0T + 5.06e4T^{2} \) |
| 41 | \( 1 - 273.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 375.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 266.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 197.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 594.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 807.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 530.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 85.8T + 3.57e5T^{2} \) |
| 73 | \( 1 + 463.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.25e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 141.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 508.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.25e3T + 9.12e5T^{2} \) |
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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
−8.867433632617732863485751544420, −7.81144697040793131768646944043, −7.22038024174972118316491043550, −6.46560119556234527872751103659, −5.54437467212630243330406901198, −4.57707136351012098982930489505, −3.62902157620089940221784407549, −2.78282542531429569650511002225, −1.43659437056108675197357025778, −0.43957364783193636616135185777,
0.43957364783193636616135185777, 1.43659437056108675197357025778, 2.78282542531429569650511002225, 3.62902157620089940221784407549, 4.57707136351012098982930489505, 5.54437467212630243330406901198, 6.46560119556234527872751103659, 7.22038024174972118316491043550, 7.81144697040793131768646944043, 8.867433632617732863485751544420
