| L(s) = 1 | + 2·2-s + 5.94·3-s + 4·4-s − 7.71·5-s + 11.8·6-s + 22.4·7-s + 8·8-s + 8.32·9-s − 15.4·10-s − 1.56·11-s + 23.7·12-s − 38.0·13-s + 44.8·14-s − 45.8·15-s + 16·16-s − 103.·17-s + 16.6·18-s + 121.·19-s − 30.8·20-s + 133.·21-s − 3.13·22-s + 41.1·23-s + 47.5·24-s − 65.5·25-s − 76.0·26-s − 111.·27-s + 89.6·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.14·3-s + 0.5·4-s − 0.689·5-s + 0.808·6-s + 1.21·7-s + 0.353·8-s + 0.308·9-s − 0.487·10-s − 0.0429·11-s + 0.571·12-s − 0.811·13-s + 0.855·14-s − 0.789·15-s + 0.250·16-s − 1.47·17-s + 0.217·18-s + 1.46·19-s − 0.344·20-s + 1.38·21-s − 0.0303·22-s + 0.373·23-s + 0.404·24-s − 0.524·25-s − 0.573·26-s − 0.791·27-s + 0.605·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 74 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.824035481\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.824035481\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 37 | \( 1 - 37T \) |
| good | 3 | \( 1 - 5.94T + 27T^{2} \) |
| 5 | \( 1 + 7.71T + 125T^{2} \) |
| 7 | \( 1 - 22.4T + 343T^{2} \) |
| 11 | \( 1 + 1.56T + 1.33e3T^{2} \) |
| 13 | \( 1 + 38.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 103.T + 4.91e3T^{2} \) |
| 19 | \( 1 - 121.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 41.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 235.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 35.3T + 2.97e4T^{2} \) |
| 41 | \( 1 - 32.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 190.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 393.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 312.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 35.3T + 2.05e5T^{2} \) |
| 61 | \( 1 - 209.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 267.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 255.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 955.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 1.30e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 155.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 754.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.36e3T + 9.12e5T^{2} \) |
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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
−14.14590965485504469632079286672, −13.30981039826992189313775111853, −11.85854117450678666128680558864, −11.11196406490509107105562072260, −9.333898564470110507953576634426, −8.088484147158937699201083407132, −7.29901705363922440616977942070, −5.15128604280149089752487640529, −3.83942942557648481900165550905, −2.26998011600194616729009763662,
2.26998011600194616729009763662, 3.83942942557648481900165550905, 5.15128604280149089752487640529, 7.29901705363922440616977942070, 8.088484147158937699201083407132, 9.333898564470110507953576634426, 11.11196406490509107105562072260, 11.85854117450678666128680558864, 13.30981039826992189313775111853, 14.14590965485504469632079286672
