L(s) = 1 | − 1.96·2-s + 3.10·3-s + 1.87·4-s − 6.10·6-s − 2.84·7-s + 0.255·8-s + 6.62·9-s − 0.295·11-s + 5.80·12-s + 2.61·13-s + 5.59·14-s − 4.24·16-s − 7.09·17-s − 13.0·18-s − 8.82·21-s + 0.581·22-s − 2.66·23-s + 0.793·24-s − 5.15·26-s + 11.2·27-s − 5.31·28-s − 1.25·29-s − 1.74·31-s + 7.83·32-s − 0.917·33-s + 13.9·34-s + 12.3·36-s + ⋯ |
L(s) = 1 | − 1.39·2-s + 1.79·3-s + 0.935·4-s − 2.49·6-s − 1.07·7-s + 0.0904·8-s + 2.20·9-s − 0.0891·11-s + 1.67·12-s + 0.726·13-s + 1.49·14-s − 1.06·16-s − 1.72·17-s − 3.07·18-s − 1.92·21-s + 0.124·22-s − 0.555·23-s + 0.161·24-s − 1.01·26-s + 2.16·27-s − 1.00·28-s − 0.232·29-s − 0.312·31-s + 1.38·32-s − 0.159·33-s + 2.39·34-s + 2.06·36-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.525702832\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.525702832\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + 1.96T + 2T^{2} \) |
| 3 | \( 1 - 3.10T + 3T^{2} \) |
| 7 | \( 1 + 2.84T + 7T^{2} \) |
| 11 | \( 1 + 0.295T + 11T^{2} \) |
| 13 | \( 1 - 2.61T + 13T^{2} \) |
| 17 | \( 1 + 7.09T + 17T^{2} \) |
| 23 | \( 1 + 2.66T + 23T^{2} \) |
| 29 | \( 1 + 1.25T + 29T^{2} \) |
| 31 | \( 1 + 1.74T + 31T^{2} \) |
| 37 | \( 1 - 0.722T + 37T^{2} \) |
| 41 | \( 1 - 10.1T + 41T^{2} \) |
| 43 | \( 1 + 4.02T + 43T^{2} \) |
| 47 | \( 1 + 2.94T + 47T^{2} \) |
| 53 | \( 1 - 6.98T + 53T^{2} \) |
| 59 | \( 1 + 8.84T + 59T^{2} \) |
| 61 | \( 1 - 6.62T + 61T^{2} \) |
| 67 | \( 1 + 1.93T + 67T^{2} \) |
| 71 | \( 1 - 15.7T + 71T^{2} \) |
| 73 | \( 1 - 3.05T + 73T^{2} \) |
| 79 | \( 1 - 8.06T + 79T^{2} \) |
| 83 | \( 1 - 13.8T + 83T^{2} \) |
| 89 | \( 1 - 0.551T + 89T^{2} \) |
| 97 | \( 1 - 6.60T + 97T^{2} \) |
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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
−8.115506675920980948207371717201, −7.27615023717784091707836597182, −6.78328383638200627854028249471, −6.11146501755307618303885121455, −4.65260963481661553253199303454, −3.93745229913833122182847727000, −3.27622168855904748990406382359, −2.33495886304548340400694827408, −1.90738280120434519255944896453, −0.65743654120377310005585441994,
0.65743654120377310005585441994, 1.90738280120434519255944896453, 2.33495886304548340400694827408, 3.27622168855904748990406382359, 3.93745229913833122182847727000, 4.65260963481661553253199303454, 6.11146501755307618303885121455, 6.78328383638200627854028249471, 7.27615023717784091707836597182, 8.115506675920980948207371717201