L(s) = 1 | + 1.56·2-s + 3-s + 0.438·4-s + 1.56·5-s + 1.56·6-s − 2.43·8-s + 9-s + 2.43·10-s − 2·11-s + 0.438·12-s − 6.12·13-s + 1.56·15-s − 4.68·16-s − 7.56·17-s + 1.56·18-s + 1.43·19-s + 0.684·20-s − 3.12·22-s + 23-s − 2.43·24-s − 2.56·25-s − 9.56·26-s + 27-s − 9.12·29-s + 2.43·30-s + 5.68·31-s − 2.43·32-s + ⋯ |
L(s) = 1 | + 1.10·2-s + 0.577·3-s + 0.219·4-s + 0.698·5-s + 0.637·6-s − 0.862·8-s + 0.333·9-s + 0.771·10-s − 0.603·11-s + 0.126·12-s − 1.69·13-s + 0.403·15-s − 1.17·16-s − 1.83·17-s + 0.368·18-s + 0.330·19-s + 0.153·20-s − 0.665·22-s + 0.208·23-s − 0.497·24-s − 0.512·25-s − 1.87·26-s + 0.192·27-s − 1.69·29-s + 0.445·30-s + 1.02·31-s − 0.431·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3381 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| 23 | \( 1 - T \) |
good | 2 | \( 1 - 1.56T + 2T^{2} \) |
| 5 | \( 1 - 1.56T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 6.12T + 13T^{2} \) |
| 17 | \( 1 + 7.56T + 17T^{2} \) |
| 19 | \( 1 - 1.43T + 19T^{2} \) |
| 29 | \( 1 + 9.12T + 29T^{2} \) |
| 31 | \( 1 - 5.68T + 31T^{2} \) |
| 37 | \( 1 - 3.43T + 37T^{2} \) |
| 41 | \( 1 + 10.2T + 41T^{2} \) |
| 43 | \( 1 + 0.315T + 43T^{2} \) |
| 47 | \( 1 - 6.68T + 47T^{2} \) |
| 53 | \( 1 + 7.80T + 53T^{2} \) |
| 59 | \( 1 - 9.12T + 59T^{2} \) |
| 61 | \( 1 - 6T + 61T^{2} \) |
| 67 | \( 1 + 14.1T + 67T^{2} \) |
| 71 | \( 1 - 13.8T + 71T^{2} \) |
| 73 | \( 1 - 5.87T + 73T^{2} \) |
| 79 | \( 1 + 5.43T + 79T^{2} \) |
| 83 | \( 1 - 4.87T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 - 15.3T + 97T^{2} \) |
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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
−8.251726715861770275855499810372, −7.32653951904363847218374502153, −6.66099184086376268156908147175, −5.77225578078412080225142227353, −5.01681873223129757355757135019, −4.51345740386899872100072446248, −3.55417765653414788827892534489, −2.51623208772260485246108126343, −2.13855963841063236199853621938, 0,
2.13855963841063236199853621938, 2.51623208772260485246108126343, 3.55417765653414788827892534489, 4.51345740386899872100072446248, 5.01681873223129757355757135019, 5.77225578078412080225142227353, 6.66099184086376268156908147175, 7.32653951904363847218374502153, 8.251726715861770275855499810372