| L(s) = 1 | − 1.89·3-s − 0.652·5-s − 1.12·7-s + 0.576·9-s − 6.15·11-s + 3.09·13-s + 1.23·15-s + 0.0373·17-s − 1.36·19-s + 2.13·21-s + 0.620·23-s − 4.57·25-s + 4.58·27-s − 9.36·29-s − 7.55·31-s + 11.6·33-s + 0.737·35-s + 11.0·37-s − 5.84·39-s − 6.87·41-s + 0.0441·43-s − 0.376·45-s − 6.75·47-s − 5.72·49-s − 0.0705·51-s − 4.82·53-s + 4.01·55-s + ⋯ |
| L(s) = 1 | − 1.09·3-s − 0.291·5-s − 0.427·7-s + 0.192·9-s − 1.85·11-s + 0.857·13-s + 0.318·15-s + 0.00904·17-s − 0.313·19-s + 0.466·21-s + 0.129·23-s − 0.914·25-s + 0.882·27-s − 1.73·29-s − 1.35·31-s + 2.02·33-s + 0.124·35-s + 1.81·37-s − 0.935·39-s − 1.07·41-s + 0.00673·43-s − 0.0560·45-s − 0.985·47-s − 0.817·49-s − 0.00988·51-s − 0.662·53-s + 0.541·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3701283655\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3701283655\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 251 | \( 1 + T \) |
| good | 3 | \( 1 + 1.89T + 3T^{2} \) |
| 5 | \( 1 + 0.652T + 5T^{2} \) |
| 7 | \( 1 + 1.12T + 7T^{2} \) |
| 11 | \( 1 + 6.15T + 11T^{2} \) |
| 13 | \( 1 - 3.09T + 13T^{2} \) |
| 17 | \( 1 - 0.0373T + 17T^{2} \) |
| 19 | \( 1 + 1.36T + 19T^{2} \) |
| 23 | \( 1 - 0.620T + 23T^{2} \) |
| 29 | \( 1 + 9.36T + 29T^{2} \) |
| 31 | \( 1 + 7.55T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 + 6.87T + 41T^{2} \) |
| 43 | \( 1 - 0.0441T + 43T^{2} \) |
| 47 | \( 1 + 6.75T + 47T^{2} \) |
| 53 | \( 1 + 4.82T + 53T^{2} \) |
| 59 | \( 1 - 9.92T + 59T^{2} \) |
| 61 | \( 1 + 6.81T + 61T^{2} \) |
| 67 | \( 1 - 12.3T + 67T^{2} \) |
| 71 | \( 1 + 5.52T + 71T^{2} \) |
| 73 | \( 1 - 3.97T + 73T^{2} \) |
| 79 | \( 1 + 10.1T + 79T^{2} \) |
| 83 | \( 1 - 10.2T + 83T^{2} \) |
| 89 | \( 1 + 3.89T + 89T^{2} \) |
| 97 | \( 1 - 14.2T + 97T^{2} \) |
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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
−8.221454817134826831301587128350, −7.77748167531288279365146274430, −6.88698790534128351606052082179, −6.04108972844139354028391607182, −5.55848318488537146766200797663, −4.92263480476241631437323458891, −3.87157392448094005997574594186, −3.03235446533348859791236181229, −1.90319581302790366768000042465, −0.34991876536475540370172811115,
0.34991876536475540370172811115, 1.90319581302790366768000042465, 3.03235446533348859791236181229, 3.87157392448094005997574594186, 4.92263480476241631437323458891, 5.55848318488537146766200797663, 6.04108972844139354028391607182, 6.88698790534128351606052082179, 7.77748167531288279365146274430, 8.221454817134826831301587128350
