L(s) = 1 | − 2-s − 2.50·3-s + 4-s + 5-s + 2.50·6-s − 8-s + 3.28·9-s − 10-s − 11-s − 2.50·12-s + 0.456·13-s − 2.50·15-s + 16-s − 0.542·17-s − 3.28·18-s + 8.41·19-s + 20-s + 22-s − 8.33·23-s + 2.50·24-s + 25-s − 0.456·26-s − 0.717·27-s + 10.7·29-s + 2.50·30-s − 5.54·31-s − 32-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 1.44·3-s + 0.5·4-s + 0.447·5-s + 1.02·6-s − 0.353·8-s + 1.09·9-s − 0.316·10-s − 0.301·11-s − 0.723·12-s + 0.126·13-s − 0.647·15-s + 0.250·16-s − 0.131·17-s − 0.774·18-s + 1.93·19-s + 0.223·20-s + 0.213·22-s − 1.73·23-s + 0.511·24-s + 0.200·25-s − 0.0894·26-s − 0.138·27-s + 1.99·29-s + 0.457·30-s − 0.996·31-s − 0.176·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5390 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5390 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7634616927\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7634616927\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
good | 3 | \( 1 + 2.50T + 3T^{2} \) |
| 13 | \( 1 - 0.456T + 13T^{2} \) |
| 17 | \( 1 + 0.542T + 17T^{2} \) |
| 19 | \( 1 - 8.41T + 19T^{2} \) |
| 23 | \( 1 + 8.33T + 23T^{2} \) |
| 29 | \( 1 - 10.7T + 29T^{2} \) |
| 31 | \( 1 + 5.54T + 31T^{2} \) |
| 37 | \( 1 + 1.37T + 37T^{2} \) |
| 41 | \( 1 + 7.92T + 41T^{2} \) |
| 43 | \( 1 - 4.54T + 43T^{2} \) |
| 47 | \( 1 - 3.58T + 47T^{2} \) |
| 53 | \( 1 - 0.416T + 53T^{2} \) |
| 59 | \( 1 + 2.94T + 59T^{2} \) |
| 61 | \( 1 - 15.6T + 61T^{2} \) |
| 67 | \( 1 + 8.83T + 67T^{2} \) |
| 71 | \( 1 + 16.5T + 71T^{2} \) |
| 73 | \( 1 - 9.94T + 73T^{2} \) |
| 79 | \( 1 - 12.2T + 79T^{2} \) |
| 83 | \( 1 - 3.47T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 - 3.97T + 97T^{2} \) |
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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
−8.101822676056184083400221353803, −7.40253452396240439817430220650, −6.65813351167467774886747327304, −6.05948725964015248142452830747, −5.42663351812685531433161528621, −4.85974356202334660030341062920, −3.70328912851617108777420414787, −2.61701621211682451993562995318, −1.50467573202222255381572945579, −0.59437987589695824458176678839,
0.59437987589695824458176678839, 1.50467573202222255381572945579, 2.61701621211682451993562995318, 3.70328912851617108777420414787, 4.85974356202334660030341062920, 5.42663351812685531433161528621, 6.05948725964015248142452830747, 6.65813351167467774886747327304, 7.40253452396240439817430220650, 8.101822676056184083400221353803