| L(s) = 1 | − 0.209·3-s + 1.33·5-s − 7-s − 2.95·9-s + 11-s − 13-s − 0.279·15-s + 2.31·17-s + 3.06·19-s + 0.209·21-s − 1.82·23-s − 3.20·25-s + 1.24·27-s − 5.86·29-s − 2.91·31-s − 0.209·33-s − 1.33·35-s + 9.05·37-s + 0.209·39-s − 1.84·41-s + 0.138·43-s − 3.95·45-s − 6.76·47-s + 49-s − 0.484·51-s + 8.09·53-s + 1.33·55-s + ⋯ |
| L(s) = 1 | − 0.120·3-s + 0.598·5-s − 0.377·7-s − 0.985·9-s + 0.301·11-s − 0.277·13-s − 0.0722·15-s + 0.561·17-s + 0.702·19-s + 0.0456·21-s − 0.380·23-s − 0.641·25-s + 0.239·27-s − 1.08·29-s − 0.523·31-s − 0.0363·33-s − 0.226·35-s + 1.48·37-s + 0.0334·39-s − 0.288·41-s + 0.0210·43-s − 0.589·45-s − 0.986·47-s + 0.142·49-s − 0.0677·51-s + 1.11·53-s + 0.180·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 + T \) |
| good | 3 | \( 1 + 0.209T + 3T^{2} \) |
| 5 | \( 1 - 1.33T + 5T^{2} \) |
| 17 | \( 1 - 2.31T + 17T^{2} \) |
| 19 | \( 1 - 3.06T + 19T^{2} \) |
| 23 | \( 1 + 1.82T + 23T^{2} \) |
| 29 | \( 1 + 5.86T + 29T^{2} \) |
| 31 | \( 1 + 2.91T + 31T^{2} \) |
| 37 | \( 1 - 9.05T + 37T^{2} \) |
| 41 | \( 1 + 1.84T + 41T^{2} \) |
| 43 | \( 1 - 0.138T + 43T^{2} \) |
| 47 | \( 1 + 6.76T + 47T^{2} \) |
| 53 | \( 1 - 8.09T + 53T^{2} \) |
| 59 | \( 1 - 2.21T + 59T^{2} \) |
| 61 | \( 1 + 11.6T + 61T^{2} \) |
| 67 | \( 1 + 12.8T + 67T^{2} \) |
| 71 | \( 1 - 0.472T + 71T^{2} \) |
| 73 | \( 1 + 5.79T + 73T^{2} \) |
| 79 | \( 1 - 4.19T + 79T^{2} \) |
| 83 | \( 1 + 5.30T + 83T^{2} \) |
| 89 | \( 1 + 10.3T + 89T^{2} \) |
| 97 | \( 1 + 8.30T + 97T^{2} \) |
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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
−8.011640278669845889202430274398, −7.42142886365603024789227050737, −6.45828297771633731619121102169, −5.76797843694287366093188027295, −5.38441567822445799364782407921, −4.23072700382223079836904930523, −3.31298396931668805225902990335, −2.53392617876398331635828134139, −1.45096949731917424460525107782, 0,
1.45096949731917424460525107782, 2.53392617876398331635828134139, 3.31298396931668805225902990335, 4.23072700382223079836904930523, 5.38441567822445799364782407921, 5.76797843694287366093188027295, 6.45828297771633731619121102169, 7.42142886365603024789227050737, 8.011640278669845889202430274398
