| L(s) = 1 | + 2.45·2-s + 4.00·4-s + 3.55·5-s − 0.769·7-s + 4.91·8-s + 8.71·10-s + 5.40·11-s − 2.95·13-s − 1.88·14-s + 4.03·16-s − 8.13·17-s − 2.72·19-s + 14.2·20-s + 13.2·22-s + 6.87·23-s + 7.63·25-s − 7.23·26-s − 3.08·28-s − 1.15·29-s + 10.1·31-s + 0.0518·32-s − 19.9·34-s − 2.73·35-s + 8.27·37-s − 6.66·38-s + 17.4·40-s + 3.54·41-s + ⋯ |
| L(s) = 1 | + 1.73·2-s + 2.00·4-s + 1.58·5-s − 0.290·7-s + 1.73·8-s + 2.75·10-s + 1.62·11-s − 0.818·13-s − 0.503·14-s + 1.00·16-s − 1.97·17-s − 0.624·19-s + 3.18·20-s + 2.82·22-s + 1.43·23-s + 1.52·25-s − 1.41·26-s − 0.582·28-s − 0.214·29-s + 1.81·31-s + 0.00916·32-s − 3.41·34-s − 0.462·35-s + 1.35·37-s − 1.08·38-s + 2.76·40-s + 0.552·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4023 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(7.567897626\) |
| \(L(\frac12)\) |
\(\approx\) |
\(7.567897626\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 149 | \( 1 - T \) |
| good | 2 | \( 1 - 2.45T + 2T^{2} \) |
| 5 | \( 1 - 3.55T + 5T^{2} \) |
| 7 | \( 1 + 0.769T + 7T^{2} \) |
| 11 | \( 1 - 5.40T + 11T^{2} \) |
| 13 | \( 1 + 2.95T + 13T^{2} \) |
| 17 | \( 1 + 8.13T + 17T^{2} \) |
| 19 | \( 1 + 2.72T + 19T^{2} \) |
| 23 | \( 1 - 6.87T + 23T^{2} \) |
| 29 | \( 1 + 1.15T + 29T^{2} \) |
| 31 | \( 1 - 10.1T + 31T^{2} \) |
| 37 | \( 1 - 8.27T + 37T^{2} \) |
| 41 | \( 1 - 3.54T + 41T^{2} \) |
| 43 | \( 1 - 4.30T + 43T^{2} \) |
| 47 | \( 1 - 5.42T + 47T^{2} \) |
| 53 | \( 1 + 10.4T + 53T^{2} \) |
| 59 | \( 1 + 1.20T + 59T^{2} \) |
| 61 | \( 1 - 8.45T + 61T^{2} \) |
| 67 | \( 1 - 12.6T + 67T^{2} \) |
| 71 | \( 1 + 1.70T + 71T^{2} \) |
| 73 | \( 1 + 15.0T + 73T^{2} \) |
| 79 | \( 1 + 6.79T + 79T^{2} \) |
| 83 | \( 1 + 9.19T + 83T^{2} \) |
| 89 | \( 1 + 5.62T + 89T^{2} \) |
| 97 | \( 1 + 1.92T + 97T^{2} \) |
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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
−8.573645888318826501705107884948, −7.05761248596534853199571037884, −6.56948479634147190718718842958, −6.26217053850657936188597772974, −5.44952538264123212630675505617, −4.51513762807028105683635090413, −4.23284298559310530469554685693, −2.82323255754603458526186156829, −2.42455580629115855000593006955, −1.40178789905980078198514762847,
1.40178789905980078198514762847, 2.42455580629115855000593006955, 2.82323255754603458526186156829, 4.23284298559310530469554685693, 4.51513762807028105683635090413, 5.44952538264123212630675505617, 6.26217053850657936188597772974, 6.56948479634147190718718842958, 7.05761248596534853199571037884, 8.573645888318826501705107884948
