| L(s) = 1 | − 1.95·2-s + 0.270·3-s + 1.82·4-s + 5-s − 0.530·6-s + 7-s + 0.333·8-s − 2.92·9-s − 1.95·10-s + 0.495·12-s − 0.690·13-s − 1.95·14-s + 0.270·15-s − 4.31·16-s − 3.49·17-s + 5.72·18-s − 6.97·19-s + 1.82·20-s + 0.270·21-s + 7.74·23-s + 0.0904·24-s + 25-s + 1.35·26-s − 1.60·27-s + 1.82·28-s − 4.56·29-s − 0.530·30-s + ⋯ |
| L(s) = 1 | − 1.38·2-s + 0.156·3-s + 0.914·4-s + 0.447·5-s − 0.216·6-s + 0.377·7-s + 0.117·8-s − 0.975·9-s − 0.618·10-s + 0.143·12-s − 0.191·13-s − 0.523·14-s + 0.0699·15-s − 1.07·16-s − 0.846·17-s + 1.34·18-s − 1.60·19-s + 0.409·20-s + 0.0591·21-s + 1.61·23-s + 0.0184·24-s + 0.200·25-s + 0.264·26-s − 0.309·27-s + 0.345·28-s − 0.846·29-s − 0.0968·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.7275994092\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7275994092\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 1.95T + 2T^{2} \) |
| 3 | \( 1 - 0.270T + 3T^{2} \) |
| 13 | \( 1 + 0.690T + 13T^{2} \) |
| 17 | \( 1 + 3.49T + 17T^{2} \) |
| 19 | \( 1 + 6.97T + 19T^{2} \) |
| 23 | \( 1 - 7.74T + 23T^{2} \) |
| 29 | \( 1 + 4.56T + 29T^{2} \) |
| 31 | \( 1 - 1.84T + 31T^{2} \) |
| 37 | \( 1 + 9.07T + 37T^{2} \) |
| 41 | \( 1 + 9.13T + 41T^{2} \) |
| 43 | \( 1 - 1.88T + 43T^{2} \) |
| 47 | \( 1 - 6.83T + 47T^{2} \) |
| 53 | \( 1 + 3.36T + 53T^{2} \) |
| 59 | \( 1 - 9.61T + 59T^{2} \) |
| 61 | \( 1 - 4.07T + 61T^{2} \) |
| 67 | \( 1 - 8.79T + 67T^{2} \) |
| 71 | \( 1 - 14.9T + 71T^{2} \) |
| 73 | \( 1 - 7.61T + 73T^{2} \) |
| 79 | \( 1 - 14.2T + 79T^{2} \) |
| 83 | \( 1 + 0.222T + 83T^{2} \) |
| 89 | \( 1 - 10.4T + 89T^{2} \) |
| 97 | \( 1 + 12.6T + 97T^{2} \) |
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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
−8.482285122250671463842687484153, −8.063994771247066775790382129602, −6.90723313895035075995824529897, −6.65754099213940897618589698304, −5.44036725661090022395897429243, −4.80485329453908334018241510184, −3.67864196401535401959634334835, −2.42390942450625529099630102898, −1.91355557303148436669259146243, −0.57455914434712772116560426897,
0.57455914434712772116560426897, 1.91355557303148436669259146243, 2.42390942450625529099630102898, 3.67864196401535401959634334835, 4.80485329453908334018241510184, 5.44036725661090022395897429243, 6.65754099213940897618589698304, 6.90723313895035075995824529897, 8.063994771247066775790382129602, 8.482285122250671463842687484153
