| L(s) = 1 | − 2-s + 1.26·3-s + 4-s − 5-s − 1.26·6-s − 2.46·7-s − 8-s − 1.41·9-s + 10-s + 5.06·11-s + 1.26·12-s + 4.28·13-s + 2.46·14-s − 1.26·15-s + 16-s − 1.55·17-s + 1.41·18-s − 4.18·19-s − 20-s − 3.10·21-s − 5.06·22-s − 0.953·23-s − 1.26·24-s + 25-s − 4.28·26-s − 5.55·27-s − 2.46·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.727·3-s + 0.5·4-s − 0.447·5-s − 0.514·6-s − 0.931·7-s − 0.353·8-s − 0.470·9-s + 0.316·10-s + 1.52·11-s + 0.363·12-s + 1.18·13-s + 0.658·14-s − 0.325·15-s + 0.250·16-s − 0.376·17-s + 0.332·18-s − 0.960·19-s − 0.223·20-s − 0.677·21-s − 1.08·22-s − 0.198·23-s − 0.257·24-s + 0.200·25-s − 0.839·26-s − 1.06·27-s − 0.465·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 601 | \( 1 + T \) |
| good | 3 | \( 1 - 1.26T + 3T^{2} \) |
| 7 | \( 1 + 2.46T + 7T^{2} \) |
| 11 | \( 1 - 5.06T + 11T^{2} \) |
| 13 | \( 1 - 4.28T + 13T^{2} \) |
| 17 | \( 1 + 1.55T + 17T^{2} \) |
| 19 | \( 1 + 4.18T + 19T^{2} \) |
| 23 | \( 1 + 0.953T + 23T^{2} \) |
| 29 | \( 1 - 8.67T + 29T^{2} \) |
| 31 | \( 1 + 3.05T + 31T^{2} \) |
| 37 | \( 1 + 8.95T + 37T^{2} \) |
| 41 | \( 1 - 10.5T + 41T^{2} \) |
| 43 | \( 1 + 11.6T + 43T^{2} \) |
| 47 | \( 1 + 11.4T + 47T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + 12.2T + 59T^{2} \) |
| 61 | \( 1 + 2.75T + 61T^{2} \) |
| 67 | \( 1 + 3.16T + 67T^{2} \) |
| 71 | \( 1 - 3.53T + 71T^{2} \) |
| 73 | \( 1 - 12.9T + 73T^{2} \) |
| 79 | \( 1 - 0.447T + 79T^{2} \) |
| 83 | \( 1 + 1.93T + 83T^{2} \) |
| 89 | \( 1 - 1.81T + 89T^{2} \) |
| 97 | \( 1 - 15.2T + 97T^{2} \) |
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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
−8.042136144732102572253627163873, −6.94722208785956184630261736407, −6.46368228465179500007432146251, −6.01049307371515985278504712942, −4.64064447361778863246992146707, −3.57259309110671341482848140992, −3.42576953023211958582393904514, −2.27732396715075525571222084377, −1.28725769965791831493958231421, 0,
1.28725769965791831493958231421, 2.27732396715075525571222084377, 3.42576953023211958582393904514, 3.57259309110671341482848140992, 4.64064447361778863246992146707, 6.01049307371515985278504712942, 6.46368228465179500007432146251, 6.94722208785956184630261736407, 8.042136144732102572253627163873
