| L(s) = 1 | − 0.308·3-s − 1.98·5-s − 2.43·7-s − 2.90·9-s − 4.25·11-s − 4.90·13-s + 0.612·15-s + 1.89·17-s − 6.15·19-s + 0.750·21-s − 1.05·25-s + 1.82·27-s + 8.05·29-s − 6.96·31-s + 1.31·33-s + 4.83·35-s − 5.61·37-s + 1.51·39-s − 3.43·41-s − 8.82·43-s + 5.76·45-s − 2.13·47-s − 1.08·49-s − 0.585·51-s − 7.26·53-s + 8.44·55-s + 1.89·57-s + ⋯ |
| L(s) = 1 | − 0.178·3-s − 0.888·5-s − 0.919·7-s − 0.968·9-s − 1.28·11-s − 1.36·13-s + 0.158·15-s + 0.460·17-s − 1.41·19-s + 0.163·21-s − 0.211·25-s + 0.350·27-s + 1.49·29-s − 1.25·31-s + 0.228·33-s + 0.816·35-s − 0.923·37-s + 0.242·39-s − 0.537·41-s − 1.34·43-s + 0.859·45-s − 0.310·47-s − 0.154·49-s − 0.0820·51-s − 0.998·53-s + 1.13·55-s + 0.251·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4232 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.04744397618\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04744397618\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + 0.308T + 3T^{2} \) |
| 5 | \( 1 + 1.98T + 5T^{2} \) |
| 7 | \( 1 + 2.43T + 7T^{2} \) |
| 11 | \( 1 + 4.25T + 11T^{2} \) |
| 13 | \( 1 + 4.90T + 13T^{2} \) |
| 17 | \( 1 - 1.89T + 17T^{2} \) |
| 19 | \( 1 + 6.15T + 19T^{2} \) |
| 29 | \( 1 - 8.05T + 29T^{2} \) |
| 31 | \( 1 + 6.96T + 31T^{2} \) |
| 37 | \( 1 + 5.61T + 37T^{2} \) |
| 41 | \( 1 + 3.43T + 41T^{2} \) |
| 43 | \( 1 + 8.82T + 43T^{2} \) |
| 47 | \( 1 + 2.13T + 47T^{2} \) |
| 53 | \( 1 + 7.26T + 53T^{2} \) |
| 59 | \( 1 - 2.65T + 59T^{2} \) |
| 61 | \( 1 + 10.3T + 61T^{2} \) |
| 67 | \( 1 - 3.71T + 67T^{2} \) |
| 71 | \( 1 - 10.9T + 71T^{2} \) |
| 73 | \( 1 - 9.61T + 73T^{2} \) |
| 79 | \( 1 - 8.50T + 79T^{2} \) |
| 83 | \( 1 - 1.89T + 83T^{2} \) |
| 89 | \( 1 + 12.1T + 89T^{2} \) |
| 97 | \( 1 - 13.0T + 97T^{2} \) |
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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
−8.170571831689828242821577756111, −7.86706841353063872689320325463, −6.88215422120808861172530611280, −6.32038301340090204145087883753, −5.26468026189649818496093896589, −4.83721706221169752892863409835, −3.65822874770165751228000681934, −2.99381995661717721347003430843, −2.19183798646785148357376617879, −0.11301769854140592958173141172,
0.11301769854140592958173141172, 2.19183798646785148357376617879, 2.99381995661717721347003430843, 3.65822874770165751228000681934, 4.83721706221169752892863409835, 5.26468026189649818496093896589, 6.32038301340090204145087883753, 6.88215422120808861172530611280, 7.86706841353063872689320325463, 8.170571831689828242821577756111
