| L(s) = 1 | − 2.56·2-s + 1.79·3-s + 4.56·4-s − 4.59·6-s − 3.80·7-s − 6.57·8-s + 0.213·9-s + 8.18·12-s + 4.49·13-s + 9.73·14-s + 7.71·16-s − 5.01·17-s − 0.546·18-s + 3.46·19-s − 6.81·21-s + 0.105·23-s − 11.7·24-s − 11.5·26-s − 4.99·27-s − 17.3·28-s + 2.35·29-s + 4.74·31-s − 6.62·32-s + 12.8·34-s + 0.974·36-s − 5.60·37-s − 8.86·38-s + ⋯ |
| L(s) = 1 | − 1.81·2-s + 1.03·3-s + 2.28·4-s − 1.87·6-s − 1.43·7-s − 2.32·8-s + 0.0711·9-s + 2.36·12-s + 1.24·13-s + 2.60·14-s + 1.92·16-s − 1.21·17-s − 0.128·18-s + 0.793·19-s − 1.48·21-s + 0.0220·23-s − 2.40·24-s − 2.26·26-s − 0.961·27-s − 3.27·28-s + 0.436·29-s + 0.852·31-s − 1.17·32-s + 2.20·34-s + 0.162·36-s − 0.920·37-s − 1.43·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.56T + 2T^{2} \) |
| 3 | \( 1 - 1.79T + 3T^{2} \) |
| 7 | \( 1 + 3.80T + 7T^{2} \) |
| 13 | \( 1 - 4.49T + 13T^{2} \) |
| 17 | \( 1 + 5.01T + 17T^{2} \) |
| 19 | \( 1 - 3.46T + 19T^{2} \) |
| 23 | \( 1 - 0.105T + 23T^{2} \) |
| 29 | \( 1 - 2.35T + 29T^{2} \) |
| 31 | \( 1 - 4.74T + 31T^{2} \) |
| 37 | \( 1 + 5.60T + 37T^{2} \) |
| 41 | \( 1 + 0.916T + 41T^{2} \) |
| 43 | \( 1 - 4.46T + 43T^{2} \) |
| 47 | \( 1 - 1.25T + 47T^{2} \) |
| 53 | \( 1 + 5.49T + 53T^{2} \) |
| 59 | \( 1 + 1.18T + 59T^{2} \) |
| 61 | \( 1 - 7.32T + 61T^{2} \) |
| 67 | \( 1 - 7.84T + 67T^{2} \) |
| 71 | \( 1 + 2.18T + 71T^{2} \) |
| 73 | \( 1 + 8.95T + 73T^{2} \) |
| 79 | \( 1 + 14.3T + 79T^{2} \) |
| 83 | \( 1 - 4.71T + 83T^{2} \) |
| 89 | \( 1 + 0.172T + 89T^{2} \) |
| 97 | \( 1 + 4.46T + 97T^{2} \) |
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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
−8.493973535060783093933365933626, −7.991150348508871059590672508422, −6.96542443497192311443945319054, −6.57027046689936331359406513460, −5.72261104021357533323795076133, −3.97189774625691580437179877383, −3.07030717124105928234241326461, −2.51755699921085063187152040890, −1.32114763052929827219137198337, 0,
1.32114763052929827219137198337, 2.51755699921085063187152040890, 3.07030717124105928234241326461, 3.97189774625691580437179877383, 5.72261104021357533323795076133, 6.57027046689936331359406513460, 6.96542443497192311443945319054, 7.991150348508871059590672508422, 8.493973535060783093933365933626
