| L(s) = 1 | + 1.78·3-s − 4.82·7-s + 0.170·9-s − 0.644·11-s + 2.99·13-s + 1.04·17-s + 0.514·19-s − 8.58·21-s + 5.48·23-s − 5.03·27-s + 4.98·29-s − 10.1·31-s − 1.14·33-s + 8.08·37-s + 5.33·39-s − 0.252·41-s − 3.51·43-s + 12.8·47-s + 16.2·49-s + 1.85·51-s − 12.7·53-s + 0.916·57-s + 0.107·59-s − 0.869·61-s − 0.821·63-s − 11.5·67-s + 9.76·69-s + ⋯ |
| L(s) = 1 | + 1.02·3-s − 1.82·7-s + 0.0567·9-s − 0.194·11-s + 0.830·13-s + 0.252·17-s + 0.118·19-s − 1.87·21-s + 1.14·23-s − 0.969·27-s + 0.925·29-s − 1.81·31-s − 0.199·33-s + 1.32·37-s + 0.853·39-s − 0.0394·41-s − 0.536·43-s + 1.87·47-s + 2.32·49-s + 0.259·51-s − 1.75·53-s + 0.121·57-s + 0.0140·59-s − 0.111·61-s − 0.103·63-s − 1.41·67-s + 1.17·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 - 1.78T + 3T^{2} \) |
| 7 | \( 1 + 4.82T + 7T^{2} \) |
| 11 | \( 1 + 0.644T + 11T^{2} \) |
| 13 | \( 1 - 2.99T + 13T^{2} \) |
| 17 | \( 1 - 1.04T + 17T^{2} \) |
| 19 | \( 1 - 0.514T + 19T^{2} \) |
| 23 | \( 1 - 5.48T + 23T^{2} \) |
| 29 | \( 1 - 4.98T + 29T^{2} \) |
| 31 | \( 1 + 10.1T + 31T^{2} \) |
| 37 | \( 1 - 8.08T + 37T^{2} \) |
| 41 | \( 1 + 0.252T + 41T^{2} \) |
| 43 | \( 1 + 3.51T + 43T^{2} \) |
| 47 | \( 1 - 12.8T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 - 0.107T + 59T^{2} \) |
| 61 | \( 1 + 0.869T + 61T^{2} \) |
| 67 | \( 1 + 11.5T + 67T^{2} \) |
| 71 | \( 1 - 1.52T + 71T^{2} \) |
| 73 | \( 1 + 4.36T + 73T^{2} \) |
| 79 | \( 1 + 11.4T + 79T^{2} \) |
| 83 | \( 1 - 5.96T + 83T^{2} \) |
| 89 | \( 1 - 7.51T + 89T^{2} \) |
| 97 | \( 1 + 10.2T + 97T^{2} \) |
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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
−7.35416796066088636293267347678, −6.68227315018434824497049709905, −6.02018072963291210056923534302, −5.44129455498724016384186762060, −4.25341564373049989605016763140, −3.54564574625839451899232815543, −3.03344094686509173393521701050, −2.52688860297331088436659706885, −1.25124209363863678627550725929, 0,
1.25124209363863678627550725929, 2.52688860297331088436659706885, 3.03344094686509173393521701050, 3.54564574625839451899232815543, 4.25341564373049989605016763140, 5.44129455498724016384186762060, 6.02018072963291210056923534302, 6.68227315018434824497049709905, 7.35416796066088636293267347678
