| L(s) = 1 | + 3-s + 3.12·5-s + 1.48·7-s + 9-s + 4.83·11-s + 3.12·15-s + 3.32·17-s + 3.82·19-s + 1.48·21-s − 4.77·23-s + 4.79·25-s + 27-s + 3.70·29-s + 6.48·31-s + 4.83·33-s + 4.66·35-s − 4.68·37-s − 9.72·41-s − 5.75·43-s + 3.12·45-s − 7.73·47-s − 4.78·49-s + 3.32·51-s + 8.29·53-s + 15.1·55-s + 3.82·57-s − 14.4·59-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.39·5-s + 0.563·7-s + 0.333·9-s + 1.45·11-s + 0.807·15-s + 0.806·17-s + 0.876·19-s + 0.325·21-s − 0.996·23-s + 0.958·25-s + 0.192·27-s + 0.687·29-s + 1.16·31-s + 0.841·33-s + 0.787·35-s − 0.770·37-s − 1.51·41-s − 0.878·43-s + 0.466·45-s − 1.12·47-s − 0.683·49-s + 0.465·51-s + 1.13·53-s + 2.04·55-s + 0.506·57-s − 1.88·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.952781997\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.952781997\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 3.12T + 5T^{2} \) |
| 7 | \( 1 - 1.48T + 7T^{2} \) |
| 11 | \( 1 - 4.83T + 11T^{2} \) |
| 17 | \( 1 - 3.32T + 17T^{2} \) |
| 19 | \( 1 - 3.82T + 19T^{2} \) |
| 23 | \( 1 + 4.77T + 23T^{2} \) |
| 29 | \( 1 - 3.70T + 29T^{2} \) |
| 31 | \( 1 - 6.48T + 31T^{2} \) |
| 37 | \( 1 + 4.68T + 37T^{2} \) |
| 41 | \( 1 + 9.72T + 41T^{2} \) |
| 43 | \( 1 + 5.75T + 43T^{2} \) |
| 47 | \( 1 + 7.73T + 47T^{2} \) |
| 53 | \( 1 - 8.29T + 53T^{2} \) |
| 59 | \( 1 + 14.4T + 59T^{2} \) |
| 61 | \( 1 + 7.18T + 61T^{2} \) |
| 67 | \( 1 + 3.81T + 67T^{2} \) |
| 71 | \( 1 - 5.09T + 71T^{2} \) |
| 73 | \( 1 - 15.7T + 73T^{2} \) |
| 79 | \( 1 - 11.1T + 79T^{2} \) |
| 83 | \( 1 - 4.06T + 83T^{2} \) |
| 89 | \( 1 + 16.9T + 89T^{2} \) |
| 97 | \( 1 + 16.9T + 97T^{2} \) |
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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
−8.425405726635224762378567514929, −7.907974361764633723514324622450, −6.75713523143090380777763311674, −6.38209886215710097849180747886, −5.43998721104795144363780061890, −4.76483926096363674659029009701, −3.72160260637961673866875597164, −2.91442172842990498748541573147, −1.75585986922911936014061244209, −1.33437374312901673127481929920,
1.33437374312901673127481929920, 1.75585986922911936014061244209, 2.91442172842990498748541573147, 3.72160260637961673866875597164, 4.76483926096363674659029009701, 5.43998721104795144363780061890, 6.38209886215710097849180747886, 6.75713523143090380777763311674, 7.907974361764633723514324622450, 8.425405726635224762378567514929
