| L(s) = 1 | + 2.50·3-s + 5-s − 3.62·7-s + 3.28·9-s + 4.89·11-s + 3.57·13-s + 2.50·15-s − 3.35·17-s + 5.34·19-s − 9.07·21-s + 0.686·23-s + 25-s + 0.703·27-s + 29-s + 2.61·31-s + 12.2·33-s − 3.62·35-s − 9.72·37-s + 8.96·39-s + 9.24·41-s − 1.35·43-s + 3.28·45-s + 9.30·47-s + 6.12·49-s − 8.40·51-s − 6.87·53-s + 4.89·55-s + ⋯ |
| L(s) = 1 | + 1.44·3-s + 0.447·5-s − 1.36·7-s + 1.09·9-s + 1.47·11-s + 0.992·13-s + 0.647·15-s − 0.812·17-s + 1.22·19-s − 1.98·21-s + 0.143·23-s + 0.200·25-s + 0.135·27-s + 0.185·29-s + 0.468·31-s + 2.13·33-s − 0.612·35-s − 1.59·37-s + 1.43·39-s + 1.44·41-s − 0.206·43-s + 0.489·45-s + 1.35·47-s + 0.875·49-s − 1.17·51-s − 0.944·53-s + 0.659·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1160 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.810772024\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.810772024\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 - 2.50T + 3T^{2} \) |
| 7 | \( 1 + 3.62T + 7T^{2} \) |
| 11 | \( 1 - 4.89T + 11T^{2} \) |
| 13 | \( 1 - 3.57T + 13T^{2} \) |
| 17 | \( 1 + 3.35T + 17T^{2} \) |
| 19 | \( 1 - 5.34T + 19T^{2} \) |
| 23 | \( 1 - 0.686T + 23T^{2} \) |
| 31 | \( 1 - 2.61T + 31T^{2} \) |
| 37 | \( 1 + 9.72T + 37T^{2} \) |
| 41 | \( 1 - 9.24T + 41T^{2} \) |
| 43 | \( 1 + 1.35T + 43T^{2} \) |
| 47 | \( 1 - 9.30T + 47T^{2} \) |
| 53 | \( 1 + 6.87T + 53T^{2} \) |
| 59 | \( 1 - 8.20T + 59T^{2} \) |
| 61 | \( 1 - 3.12T + 61T^{2} \) |
| 67 | \( 1 + 8.13T + 67T^{2} \) |
| 71 | \( 1 + 10.0T + 71T^{2} \) |
| 73 | \( 1 - 8.45T + 73T^{2} \) |
| 79 | \( 1 + 13.7T + 79T^{2} \) |
| 83 | \( 1 + 0.440T + 83T^{2} \) |
| 89 | \( 1 + 2.10T + 89T^{2} \) |
| 97 | \( 1 + 17.8T + 97T^{2} \) |
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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
−9.441395053243192584804089932914, −9.097818753967513836462786708121, −8.468022989301487420147213917506, −7.23921022252495457646042923486, −6.59865439592043696740182799361, −5.76821264033540962733404095086, −4.14464856005133557645913854901, −3.46692167029382412385997138962, −2.68224974555170966255277256742, −1.35842648165793977461904761604,
1.35842648165793977461904761604, 2.68224974555170966255277256742, 3.46692167029382412385997138962, 4.14464856005133557645913854901, 5.76821264033540962733404095086, 6.59865439592043696740182799361, 7.23921022252495457646042923486, 8.468022989301487420147213917506, 9.097818753967513836462786708121, 9.441395053243192584804089932914
