| L(s) = 1 | + 0.601·3-s − 4.19·5-s + 7-s − 2.63·9-s + 2.71·11-s + 5.83·13-s − 2.52·15-s − 17-s − 6.47·19-s + 0.601·21-s − 3.86·23-s + 12.6·25-s − 3.39·27-s + 0.714·29-s + 9.96·31-s + 1.63·33-s − 4.19·35-s + 7.80·37-s + 3.51·39-s − 3.82·41-s − 0.681·43-s + 11.0·45-s − 7.53·47-s + 49-s − 0.601·51-s − 7.25·53-s − 11.4·55-s + ⋯ |
| L(s) = 1 | + 0.347·3-s − 1.87·5-s + 0.377·7-s − 0.879·9-s + 0.818·11-s + 1.61·13-s − 0.652·15-s − 0.242·17-s − 1.48·19-s + 0.131·21-s − 0.806·23-s + 2.52·25-s − 0.652·27-s + 0.132·29-s + 1.79·31-s + 0.284·33-s − 0.709·35-s + 1.28·37-s + 0.562·39-s − 0.597·41-s − 0.103·43-s + 1.65·45-s − 1.09·47-s + 0.142·49-s − 0.0842·51-s − 0.996·53-s − 1.53·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3808 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 3 | \( 1 - 0.601T + 3T^{2} \) |
| 5 | \( 1 + 4.19T + 5T^{2} \) |
| 11 | \( 1 - 2.71T + 11T^{2} \) |
| 13 | \( 1 - 5.83T + 13T^{2} \) |
| 19 | \( 1 + 6.47T + 19T^{2} \) |
| 23 | \( 1 + 3.86T + 23T^{2} \) |
| 29 | \( 1 - 0.714T + 29T^{2} \) |
| 31 | \( 1 - 9.96T + 31T^{2} \) |
| 37 | \( 1 - 7.80T + 37T^{2} \) |
| 41 | \( 1 + 3.82T + 41T^{2} \) |
| 43 | \( 1 + 0.681T + 43T^{2} \) |
| 47 | \( 1 + 7.53T + 47T^{2} \) |
| 53 | \( 1 + 7.25T + 53T^{2} \) |
| 59 | \( 1 - 1.29T + 59T^{2} \) |
| 61 | \( 1 + 11.1T + 61T^{2} \) |
| 67 | \( 1 + 12.4T + 67T^{2} \) |
| 71 | \( 1 - 6.76T + 71T^{2} \) |
| 73 | \( 1 - 1.60T + 73T^{2} \) |
| 79 | \( 1 - 16.3T + 79T^{2} \) |
| 83 | \( 1 + 8.58T + 83T^{2} \) |
| 89 | \( 1 + 9.67T + 89T^{2} \) |
| 97 | \( 1 - 9.18T + 97T^{2} \) |
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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
−8.228853074205073251219128151348, −7.75144612950319980520769914595, −6.43453547020539655302226344647, −6.29482418253187832587092394923, −4.79010130937918531384185884124, −4.12243645778485355390197057914, −3.61581256567545850726217834206, −2.71870270576086624990711678566, −1.29517144764062230551391095014, 0,
1.29517144764062230551391095014, 2.71870270576086624990711678566, 3.61581256567545850726217834206, 4.12243645778485355390197057914, 4.79010130937918531384185884124, 6.29482418253187832587092394923, 6.43453547020539655302226344647, 7.75144612950319980520769914595, 8.228853074205073251219128151348
