L(s) = 1 | − 2-s − 2.71·3-s + 4-s − 2.38·5-s + 2.71·6-s − 4.13·7-s − 8-s + 4.34·9-s + 2.38·10-s + 5.88·11-s − 2.71·12-s − 3.00·13-s + 4.13·14-s + 6.46·15-s + 16-s + 6.38·17-s − 4.34·18-s + 4.95·19-s − 2.38·20-s + 11.2·21-s − 5.88·22-s + 23-s + 2.71·24-s + 0.691·25-s + 3.00·26-s − 3.65·27-s − 4.13·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.56·3-s + 0.5·4-s − 1.06·5-s + 1.10·6-s − 1.56·7-s − 0.353·8-s + 1.44·9-s + 0.754·10-s + 1.77·11-s − 0.782·12-s − 0.833·13-s + 1.10·14-s + 1.66·15-s + 0.250·16-s + 1.54·17-s − 1.02·18-s + 1.13·19-s − 0.533·20-s + 2.44·21-s − 1.25·22-s + 0.208·23-s + 0.553·24-s + 0.138·25-s + 0.589·26-s − 0.703·27-s − 0.782·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6026 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.5248822194\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5248822194\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 131 | \( 1 + T \) |
good | 3 | \( 1 + 2.71T + 3T^{2} \) |
| 5 | \( 1 + 2.38T + 5T^{2} \) |
| 7 | \( 1 + 4.13T + 7T^{2} \) |
| 11 | \( 1 - 5.88T + 11T^{2} \) |
| 13 | \( 1 + 3.00T + 13T^{2} \) |
| 17 | \( 1 - 6.38T + 17T^{2} \) |
| 19 | \( 1 - 4.95T + 19T^{2} \) |
| 29 | \( 1 - 9.90T + 29T^{2} \) |
| 31 | \( 1 + 5.84T + 31T^{2} \) |
| 37 | \( 1 + 5.08T + 37T^{2} \) |
| 41 | \( 1 - 11.8T + 41T^{2} \) |
| 43 | \( 1 - 4.90T + 43T^{2} \) |
| 47 | \( 1 - 7.91T + 47T^{2} \) |
| 53 | \( 1 + 6.91T + 53T^{2} \) |
| 59 | \( 1 - 2.15T + 59T^{2} \) |
| 61 | \( 1 - 1.81T + 61T^{2} \) |
| 67 | \( 1 + 3.88T + 67T^{2} \) |
| 71 | \( 1 - 12.1T + 71T^{2} \) |
| 73 | \( 1 - 6.84T + 73T^{2} \) |
| 79 | \( 1 + 15.0T + 79T^{2} \) |
| 83 | \( 1 + 3.32T + 83T^{2} \) |
| 89 | \( 1 + 6.85T + 89T^{2} \) |
| 97 | \( 1 + 12.2T + 97T^{2} \) |
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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
−7.86093872572732452106358430274, −7.08416310289336908811922749204, −6.88533921538062245788099113267, −6.00140874967013510626753815384, −5.53002112781936393727378794438, −4.39521653951707869551193479432, −3.66673035763273032341702385361, −2.93391240548889786077551453549, −1.14971110015921201042484740747, −0.56450619043106361714706016307,
0.56450619043106361714706016307, 1.14971110015921201042484740747, 2.93391240548889786077551453549, 3.66673035763273032341702385361, 4.39521653951707869551193479432, 5.53002112781936393727378794438, 6.00140874967013510626753815384, 6.88533921538062245788099113267, 7.08416310289336908811922749204, 7.86093872572732452106358430274