| L(s) = 1 | + 2·2-s + 10.0·3-s + 4·4-s + 8.61·5-s + 20.0·6-s − 13.5·7-s + 8·8-s + 73.0·9-s + 17.2·10-s + 28.7·11-s + 40.0·12-s + 57.0·13-s − 27.1·14-s + 86.1·15-s + 16·16-s − 79.9·17-s + 146.·18-s − 123.·19-s + 34.4·20-s − 135.·21-s + 57.5·22-s − 32.1·23-s + 80.0·24-s − 50.7·25-s + 114.·26-s + 460.·27-s − 54.3·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.92·3-s + 0.5·4-s + 0.770·5-s + 1.36·6-s − 0.733·7-s + 0.353·8-s + 2.70·9-s + 0.544·10-s + 0.789·11-s + 0.962·12-s + 1.21·13-s − 0.518·14-s + 1.48·15-s + 0.250·16-s − 1.14·17-s + 1.91·18-s − 1.49·19-s + 0.385·20-s − 1.41·21-s + 0.558·22-s − 0.291·23-s + 0.680·24-s − 0.406·25-s + 0.860·26-s + 3.27·27-s − 0.366·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(6.755431605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(6.755431605\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 269 | \( 1 - 269T \) |
| good | 3 | \( 1 - 10.0T + 27T^{2} \) |
| 5 | \( 1 - 8.61T + 125T^{2} \) |
| 7 | \( 1 + 13.5T + 343T^{2} \) |
| 11 | \( 1 - 28.7T + 1.33e3T^{2} \) |
| 13 | \( 1 - 57.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 79.9T + 4.91e3T^{2} \) |
| 19 | \( 1 + 123.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 32.1T + 1.21e4T^{2} \) |
| 29 | \( 1 + 267.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 208.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 150.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 326.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 111.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 133.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 356.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 303.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 567.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 40.4T + 3.00e5T^{2} \) |
| 71 | \( 1 + 222.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 816.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 419.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 238.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.07e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.20e3T + 9.12e5T^{2} \) |
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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
−10.25349858252450358456894296230, −9.134618760643616249972060024343, −8.996757049523995618693885163087, −7.74689152132039213379466816284, −6.70709839665970894271122033566, −5.99152921776377006773536980692, −4.12012795841667039327487157973, −3.74563845988830867774497259019, −2.44068385508487015405874461248, −1.72318425283666642663390960677,
1.72318425283666642663390960677, 2.44068385508487015405874461248, 3.74563845988830867774497259019, 4.12012795841667039327487157973, 5.99152921776377006773536980692, 6.70709839665970894271122033566, 7.74689152132039213379466816284, 8.996757049523995618693885163087, 9.134618760643616249972060024343, 10.25349858252450358456894296230
