| L(s) = 1 | + 1.41·3-s − 5-s − 1.41·7-s − 0.999·9-s − 2.82·11-s − 2·13-s − 1.41·15-s + 2·17-s + 5.65·19-s − 2.00·21-s − 1.41·23-s + 25-s − 5.65·27-s − 8·29-s − 8.48·31-s − 4.00·33-s + 1.41·35-s − 10·37-s − 2.82·39-s − 7.07·43-s + 0.999·45-s + 9.89·47-s − 5·49-s + 2.82·51-s − 2·53-s + 2.82·55-s + 8.00·57-s + ⋯ |
| L(s) = 1 | + 0.816·3-s − 0.447·5-s − 0.534·7-s − 0.333·9-s − 0.852·11-s − 0.554·13-s − 0.365·15-s + 0.485·17-s + 1.29·19-s − 0.436·21-s − 0.294·23-s + 0.200·25-s − 1.08·27-s − 1.48·29-s − 1.52·31-s − 0.696·33-s + 0.239·35-s − 1.64·37-s − 0.452·39-s − 1.07·43-s + 0.149·45-s + 1.44·47-s − 0.714·49-s + 0.396·51-s − 0.274·53-s + 0.381·55-s + 1.05·57-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| good | 3 | \( 1 - 1.41T + 3T^{2} \) |
| 7 | \( 1 + 1.41T + 7T^{2} \) |
| 11 | \( 1 + 2.82T + 11T^{2} \) |
| 13 | \( 1 + 2T + 13T^{2} \) |
| 17 | \( 1 - 2T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 1.41T + 23T^{2} \) |
| 29 | \( 1 + 8T + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 + 7.07T + 43T^{2} \) |
| 47 | \( 1 - 9.89T + 47T^{2} \) |
| 53 | \( 1 + 2T + 53T^{2} \) |
| 59 | \( 1 - 5.65T + 59T^{2} \) |
| 61 | \( 1 + 10T + 61T^{2} \) |
| 67 | \( 1 - 1.41T + 67T^{2} \) |
| 71 | \( 1 - 14.1T + 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 11.3T + 79T^{2} \) |
| 83 | \( 1 + 12.7T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 14T + 97T^{2} \) |
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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
−9.274962123719370167595050729223, −8.440949587671073144613569020513, −7.59187347490131035868864355647, −7.18362482506338610184882294883, −5.74443531892471900836219052439, −5.12170474264707469370700319816, −3.64206811922499903686657891385, −3.17164652379146156241944432261, −2.01795677224680399455924721736, 0,
2.01795677224680399455924721736, 3.17164652379146156241944432261, 3.64206811922499903686657891385, 5.12170474264707469370700319816, 5.74443531892471900836219052439, 7.18362482506338610184882294883, 7.59187347490131035868864355647, 8.440949587671073144613569020513, 9.274962123719370167595050729223
