L(s) = 1 | − 2-s − 0.916·3-s + 4-s − 1.80·5-s + 0.916·6-s − 2.30·7-s − 8-s − 2.15·9-s + 1.80·10-s + 5.03·11-s − 0.916·12-s + 3.75·13-s + 2.30·14-s + 1.65·15-s + 16-s − 1.14·17-s + 2.15·18-s − 19-s − 1.80·20-s + 2.10·21-s − 5.03·22-s − 3.89·23-s + 0.916·24-s − 1.74·25-s − 3.75·26-s + 4.72·27-s − 2.30·28-s + ⋯ |
L(s) = 1 | − 0.707·2-s − 0.529·3-s + 0.5·4-s − 0.806·5-s + 0.374·6-s − 0.869·7-s − 0.353·8-s − 0.719·9-s + 0.570·10-s + 1.51·11-s − 0.264·12-s + 1.04·13-s + 0.614·14-s + 0.426·15-s + 0.250·16-s − 0.277·17-s + 0.509·18-s − 0.229·19-s − 0.403·20-s + 0.460·21-s − 1.07·22-s − 0.812·23-s + 0.187·24-s − 0.349·25-s − 0.737·26-s + 0.910·27-s − 0.434·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8018 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 211 | \( 1 + T \) |
good | 3 | \( 1 + 0.916T + 3T^{2} \) |
| 5 | \( 1 + 1.80T + 5T^{2} \) |
| 7 | \( 1 + 2.30T + 7T^{2} \) |
| 11 | \( 1 - 5.03T + 11T^{2} \) |
| 13 | \( 1 - 3.75T + 13T^{2} \) |
| 17 | \( 1 + 1.14T + 17T^{2} \) |
| 23 | \( 1 + 3.89T + 23T^{2} \) |
| 29 | \( 1 + 6.08T + 29T^{2} \) |
| 31 | \( 1 - 2.27T + 31T^{2} \) |
| 37 | \( 1 - 6.22T + 37T^{2} \) |
| 41 | \( 1 - 11.2T + 41T^{2} \) |
| 43 | \( 1 + 10.5T + 43T^{2} \) |
| 47 | \( 1 + 7.77T + 47T^{2} \) |
| 53 | \( 1 - 4.91T + 53T^{2} \) |
| 59 | \( 1 - 0.349T + 59T^{2} \) |
| 61 | \( 1 - 1.02T + 61T^{2} \) |
| 67 | \( 1 - 2.71T + 67T^{2} \) |
| 71 | \( 1 - 7.52T + 71T^{2} \) |
| 73 | \( 1 + 5.77T + 73T^{2} \) |
| 79 | \( 1 - 1.97T + 79T^{2} \) |
| 83 | \( 1 + 7.97T + 83T^{2} \) |
| 89 | \( 1 - 8.56T + 89T^{2} \) |
| 97 | \( 1 + 8.80T + 97T^{2} \) |
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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
−7.50805869715302768742182135606, −6.71412615353239819406197359190, −6.16745474146092666038139710033, −5.83676899563010780228334866995, −4.51614662168885492924164575017, −3.75071419150759034923826863787, −3.26932360662409102918583398252, −2.05114496403817907500373668502, −0.930049725226640758159756835764, 0,
0.930049725226640758159756835764, 2.05114496403817907500373668502, 3.26932360662409102918583398252, 3.75071419150759034923826863787, 4.51614662168885492924164575017, 5.83676899563010780228334866995, 6.16745474146092666038139710033, 6.71412615353239819406197359190, 7.50805869715302768742182135606