| L(s) = 1 | + 2.82·2-s + 5.98·4-s + 11.2·8-s − 3·9-s + 19.8·16-s − 8.47·18-s + 7.50·23-s − 5·25-s + 4.60·29-s + 33.5·32-s − 17.9·36-s + 8.21·37-s + 6.59·43-s + 21.2·46-s − 14.1·50-s + 1.86·53-s + 13.0·58-s + 55.0·64-s + 6.09·67-s − 9.83·71-s − 33.7·72-s + 23.1·74-s − 15.8·79-s + 9·81-s + 18.6·86-s + 44.8·92-s − 29.9·100-s + ⋯ |
| L(s) = 1 | + 1.99·2-s + 2.99·4-s + 3.97·8-s − 9-s + 4.95·16-s − 1.99·18-s + 1.56·23-s − 25-s + 0.854·29-s + 5.92·32-s − 2.99·36-s + 1.34·37-s + 1.00·43-s + 3.12·46-s − 1.99·50-s + 0.256·53-s + 1.70·58-s + 6.87·64-s + 0.744·67-s − 1.16·71-s − 3.97·72-s + 2.69·74-s − 1.77·79-s + 81-s + 2.01·86-s + 4.68·92-s − 2.99·100-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5929 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(8.235280038\) |
| \(L(\frac12)\) |
\(\approx\) |
\(8.235280038\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 - 2.82T + 2T^{2} \) |
| 3 | \( 1 + 3T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 7.50T + 23T^{2} \) |
| 29 | \( 1 - 4.60T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 8.21T + 37T^{2} \) |
| 41 | \( 1 + 41T^{2} \) |
| 43 | \( 1 - 6.59T + 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 1.86T + 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 6.09T + 67T^{2} \) |
| 71 | \( 1 + 9.83T + 71T^{2} \) |
| 73 | \( 1 + 73T^{2} \) |
| 79 | \( 1 + 15.8T + 79T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 + 97T^{2} \) |
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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
−7.75881253630905358108257165160, −7.13041446635384749901192738006, −6.34077308139470516796292032152, −5.82342553589839704838132068756, −5.18761865100732785524258757119, −4.48735199612003579283658456899, −3.74977561812583921183212001733, −2.87482793688771641540401176480, −2.46793742936473786832152673606, −1.20337548620459725403953017335,
1.20337548620459725403953017335, 2.46793742936473786832152673606, 2.87482793688771641540401176480, 3.74977561812583921183212001733, 4.48735199612003579283658456899, 5.18761865100732785524258757119, 5.82342553589839704838132068756, 6.34077308139470516796292032152, 7.13041446635384749901192738006, 7.75881253630905358108257165160
