| L(s) = 1 | − 2.15·3-s − 3.43·5-s − 3.93·7-s + 1.65·9-s − 11-s + 3.31·13-s + 7.40·15-s + 2.80·17-s + 19-s + 8.50·21-s − 6.88·23-s + 6.77·25-s + 2.90·27-s + 5.67·29-s − 2.51·31-s + 2.15·33-s + 13.5·35-s − 6.39·37-s − 7.14·39-s + 0.560·41-s + 9.40·43-s − 5.67·45-s + 12.1·47-s + 8.52·49-s − 6.05·51-s + 5.68·53-s + 3.43·55-s + ⋯ |
| L(s) = 1 | − 1.24·3-s − 1.53·5-s − 1.48·7-s + 0.551·9-s − 0.301·11-s + 0.918·13-s + 1.91·15-s + 0.680·17-s + 0.229·19-s + 1.85·21-s − 1.43·23-s + 1.35·25-s + 0.558·27-s + 1.05·29-s − 0.452·31-s + 0.375·33-s + 2.28·35-s − 1.05·37-s − 1.14·39-s + 0.0875·41-s + 1.43·43-s − 0.846·45-s + 1.77·47-s + 1.21·49-s − 0.847·51-s + 0.780·53-s + 0.462·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3344 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 + 2.15T + 3T^{2} \) |
| 5 | \( 1 + 3.43T + 5T^{2} \) |
| 7 | \( 1 + 3.93T + 7T^{2} \) |
| 13 | \( 1 - 3.31T + 13T^{2} \) |
| 17 | \( 1 - 2.80T + 17T^{2} \) |
| 23 | \( 1 + 6.88T + 23T^{2} \) |
| 29 | \( 1 - 5.67T + 29T^{2} \) |
| 31 | \( 1 + 2.51T + 31T^{2} \) |
| 37 | \( 1 + 6.39T + 37T^{2} \) |
| 41 | \( 1 - 0.560T + 41T^{2} \) |
| 43 | \( 1 - 9.40T + 43T^{2} \) |
| 47 | \( 1 - 12.1T + 47T^{2} \) |
| 53 | \( 1 - 5.68T + 53T^{2} \) |
| 59 | \( 1 + 4.35T + 59T^{2} \) |
| 61 | \( 1 + 3.56T + 61T^{2} \) |
| 67 | \( 1 - 9.95T + 67T^{2} \) |
| 71 | \( 1 - 11.4T + 71T^{2} \) |
| 73 | \( 1 + 8.95T + 73T^{2} \) |
| 79 | \( 1 + 8.49T + 79T^{2} \) |
| 83 | \( 1 - 5.21T + 83T^{2} \) |
| 89 | \( 1 + 7.28T + 89T^{2} \) |
| 97 | \( 1 - 10.6T + 97T^{2} \) |
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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
−8.162246846669812964411968148405, −7.38239715980111212133284836338, −6.68776078322534258084166643027, −5.98522750860643598549821142926, −5.40220955246403025801526722449, −4.19752687054131412850554903692, −3.71309944073817472794475725733, −2.81731398990280498041084308796, −0.867345782299395195071095357339, 0,
0.867345782299395195071095357339, 2.81731398990280498041084308796, 3.71309944073817472794475725733, 4.19752687054131412850554903692, 5.40220955246403025801526722449, 5.98522750860643598549821142926, 6.68776078322534258084166643027, 7.38239715980111212133284836338, 8.162246846669812964411968148405
