L(s) = 1 | + 1.18·2-s + 3-s − 0.589·4-s + 3.30·5-s + 1.18·6-s + 1.53·7-s − 3.07·8-s + 9-s + 3.91·10-s + 5.55·11-s − 0.589·12-s + 13-s + 1.82·14-s + 3.30·15-s − 2.47·16-s − 2.82·17-s + 1.18·18-s + 1.07·19-s − 1.94·20-s + 1.53·21-s + 6.60·22-s − 2.30·23-s − 3.07·24-s + 5.89·25-s + 1.18·26-s + 27-s − 0.904·28-s + ⋯ |
L(s) = 1 | + 0.839·2-s + 0.577·3-s − 0.294·4-s + 1.47·5-s + 0.484·6-s + 0.579·7-s − 1.08·8-s + 0.333·9-s + 1.23·10-s + 1.67·11-s − 0.170·12-s + 0.277·13-s + 0.487·14-s + 0.852·15-s − 0.618·16-s − 0.685·17-s + 0.279·18-s + 0.245·19-s − 0.434·20-s + 0.334·21-s + 1.40·22-s − 0.481·23-s − 0.627·24-s + 1.17·25-s + 0.232·26-s + 0.192·27-s − 0.170·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4017 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(4.886737649\) |
\(L(\frac12)\) |
\(\approx\) |
\(4.886737649\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 103 | \( 1 - T \) |
good | 2 | \( 1 - 1.18T + 2T^{2} \) |
| 5 | \( 1 - 3.30T + 5T^{2} \) |
| 7 | \( 1 - 1.53T + 7T^{2} \) |
| 11 | \( 1 - 5.55T + 11T^{2} \) |
| 17 | \( 1 + 2.82T + 17T^{2} \) |
| 19 | \( 1 - 1.07T + 19T^{2} \) |
| 23 | \( 1 + 2.30T + 23T^{2} \) |
| 29 | \( 1 + 5.84T + 29T^{2} \) |
| 31 | \( 1 - 3.69T + 31T^{2} \) |
| 37 | \( 1 - 7.77T + 37T^{2} \) |
| 41 | \( 1 + 0.611T + 41T^{2} \) |
| 43 | \( 1 - 8.51T + 43T^{2} \) |
| 47 | \( 1 - 1.59T + 47T^{2} \) |
| 53 | \( 1 - 8.75T + 53T^{2} \) |
| 59 | \( 1 + 4.74T + 59T^{2} \) |
| 61 | \( 1 + 12.4T + 61T^{2} \) |
| 67 | \( 1 + 6.68T + 67T^{2} \) |
| 71 | \( 1 + 8.98T + 71T^{2} \) |
| 73 | \( 1 - 8.58T + 73T^{2} \) |
| 79 | \( 1 + 1.83T + 79T^{2} \) |
| 83 | \( 1 + 8.52T + 83T^{2} \) |
| 89 | \( 1 - 12.3T + 89T^{2} \) |
| 97 | \( 1 + 6.02T + 97T^{2} \) |
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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
−8.704758909518061347844624624883, −7.73486968265312696725071413015, −6.64934863136658121868152054817, −6.09935655217185484811633430133, −5.53084922950260687085679310429, −4.47414422001044024595535147287, −4.07231442193455015153529050149, −3.01122461488149960487064267494, −2.08270127726809589146095074593, −1.22989005829471760676342957989,
1.22989005829471760676342957989, 2.08270127726809589146095074593, 3.01122461488149960487064267494, 4.07231442193455015153529050149, 4.47414422001044024595535147287, 5.53084922950260687085679310429, 6.09935655217185484811633430133, 6.64934863136658121868152054817, 7.73486968265312696725071413015, 8.704758909518061347844624624883