L(s) = 1 | − 1.74·2-s − 2.44·3-s + 1.05·4-s + 5-s + 4.26·6-s + 1.65·8-s + 2.95·9-s − 1.74·10-s + 11-s − 2.57·12-s + 1.88·13-s − 2.44·15-s − 4.99·16-s − 2.57·17-s − 5.17·18-s + 3.06·19-s + 1.05·20-s − 1.74·22-s − 5.83·23-s − 4.03·24-s + 25-s − 3.28·26-s + 0.0978·27-s − 2.56·29-s + 4.26·30-s + 2.84·31-s + 5.42·32-s + ⋯ |
L(s) = 1 | − 1.23·2-s − 1.40·3-s + 0.526·4-s + 0.447·5-s + 1.74·6-s + 0.584·8-s + 0.986·9-s − 0.552·10-s + 0.301·11-s − 0.742·12-s + 0.522·13-s − 0.630·15-s − 1.24·16-s − 0.623·17-s − 1.21·18-s + 0.703·19-s + 0.235·20-s − 0.372·22-s − 1.21·23-s − 0.824·24-s + 0.200·25-s − 0.645·26-s + 0.0188·27-s − 0.475·29-s + 0.778·30-s + 0.511·31-s + 0.958·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2695 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 - T \) |
good | 2 | \( 1 + 1.74T + 2T^{2} \) |
| 3 | \( 1 + 2.44T + 3T^{2} \) |
| 13 | \( 1 - 1.88T + 13T^{2} \) |
| 17 | \( 1 + 2.57T + 17T^{2} \) |
| 19 | \( 1 - 3.06T + 19T^{2} \) |
| 23 | \( 1 + 5.83T + 23T^{2} \) |
| 29 | \( 1 + 2.56T + 29T^{2} \) |
| 31 | \( 1 - 2.84T + 31T^{2} \) |
| 37 | \( 1 + 4.98T + 37T^{2} \) |
| 41 | \( 1 - 1.57T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 + 4.87T + 47T^{2} \) |
| 53 | \( 1 + 1.90T + 53T^{2} \) |
| 59 | \( 1 - 8.36T + 59T^{2} \) |
| 61 | \( 1 + 6.89T + 61T^{2} \) |
| 67 | \( 1 - 14.6T + 67T^{2} \) |
| 71 | \( 1 - 4.51T + 71T^{2} \) |
| 73 | \( 1 - 0.100T + 73T^{2} \) |
| 79 | \( 1 - 9.19T + 79T^{2} \) |
| 83 | \( 1 - 7.62T + 83T^{2} \) |
| 89 | \( 1 - 14.1T + 89T^{2} \) |
| 97 | \( 1 - 1.79T + 97T^{2} \) |
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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
−8.510058887253662348125919117173, −7.84304387883909464044149852802, −6.76625690161578289691427416124, −6.42179721337086962977629964061, −5.43438181564502171655079334286, −4.78210799781104152461638552129, −3.70790508039427940147696987916, −2.06221113142901281924639355134, −1.11653076971653297762155559847, 0,
1.11653076971653297762155559847, 2.06221113142901281924639355134, 3.70790508039427940147696987916, 4.78210799781104152461638552129, 5.43438181564502171655079334286, 6.42179721337086962977629964061, 6.76625690161578289691427416124, 7.84304387883909464044149852802, 8.510058887253662348125919117173