| L(s) = 1 | − 2-s + 2.99·3-s + 4-s + 2.01·5-s − 2.99·6-s + 1.87·7-s − 8-s + 5.94·9-s − 2.01·10-s − 11-s + 2.99·12-s + 6.42·13-s − 1.87·14-s + 6.03·15-s + 16-s − 1.14·17-s − 5.94·18-s + 3.02·19-s + 2.01·20-s + 5.61·21-s + 22-s − 3.30·23-s − 2.99·24-s − 0.929·25-s − 6.42·26-s + 8.81·27-s + 1.87·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.72·3-s + 0.5·4-s + 0.902·5-s − 1.22·6-s + 0.709·7-s − 0.353·8-s + 1.98·9-s − 0.638·10-s − 0.301·11-s + 0.863·12-s + 1.78·13-s − 0.501·14-s + 1.55·15-s + 0.250·16-s − 0.277·17-s − 1.40·18-s + 0.693·19-s + 0.451·20-s + 1.22·21-s + 0.213·22-s − 0.689·23-s − 0.610·24-s − 0.185·25-s − 1.25·26-s + 1.69·27-s + 0.354·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4334 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.826543247\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.826543247\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 197 | \( 1 - T \) |
| good | 3 | \( 1 - 2.99T + 3T^{2} \) |
| 5 | \( 1 - 2.01T + 5T^{2} \) |
| 7 | \( 1 - 1.87T + 7T^{2} \) |
| 13 | \( 1 - 6.42T + 13T^{2} \) |
| 17 | \( 1 + 1.14T + 17T^{2} \) |
| 19 | \( 1 - 3.02T + 19T^{2} \) |
| 23 | \( 1 + 3.30T + 23T^{2} \) |
| 29 | \( 1 + 5.57T + 29T^{2} \) |
| 31 | \( 1 + 6.12T + 31T^{2} \) |
| 37 | \( 1 - 11.0T + 37T^{2} \) |
| 41 | \( 1 - 7.11T + 41T^{2} \) |
| 43 | \( 1 - 4.08T + 43T^{2} \) |
| 47 | \( 1 - 0.747T + 47T^{2} \) |
| 53 | \( 1 + 3.38T + 53T^{2} \) |
| 59 | \( 1 + 7.99T + 59T^{2} \) |
| 61 | \( 1 - 3.17T + 61T^{2} \) |
| 67 | \( 1 + 4.76T + 67T^{2} \) |
| 71 | \( 1 + 11.6T + 71T^{2} \) |
| 73 | \( 1 - 3.14T + 73T^{2} \) |
| 79 | \( 1 - 4.61T + 79T^{2} \) |
| 83 | \( 1 + 4.40T + 83T^{2} \) |
| 89 | \( 1 + 17.5T + 89T^{2} \) |
| 97 | \( 1 - 3.79T + 97T^{2} \) |
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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
−8.433549178200778543774353782893, −7.79486988178635596162399324428, −7.41414019404810056762804932863, −6.19644482036593583988626841997, −5.68125156151817714697305832669, −4.33747063872372865618645012595, −3.59001587826876510173033116574, −2.68821810307986176650421924295, −1.88603114691761448378320348178, −1.31027040308729283657892289309,
1.31027040308729283657892289309, 1.88603114691761448378320348178, 2.68821810307986176650421924295, 3.59001587826876510173033116574, 4.33747063872372865618645012595, 5.68125156151817714697305832669, 6.19644482036593583988626841997, 7.41414019404810056762804932863, 7.79486988178635596162399324428, 8.433549178200778543774353782893
