| L(s) = 1 | + 3-s − 0.454·5-s + 9-s + 5.79·11-s + 5.88·13-s − 0.454·15-s − 2.90·17-s + 5.88·19-s + 2.90·23-s − 4.79·25-s + 27-s + 3.54·29-s − 4.33·31-s + 5.79·33-s + 7.70·37-s + 5.88·39-s − 9.58·41-s − 10.7·43-s − 0.454·45-s + 4.90·47-s − 2.90·51-s + 13.1·53-s − 2.63·55-s + 5.88·57-s − 1.79·59-s − 4.67·61-s − 2.67·65-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.203·5-s + 0.333·9-s + 1.74·11-s + 1.63·13-s − 0.117·15-s − 0.705·17-s + 1.34·19-s + 0.606·23-s − 0.958·25-s + 0.192·27-s + 0.658·29-s − 0.779·31-s + 1.00·33-s + 1.26·37-s + 0.942·39-s − 1.49·41-s − 1.64·43-s − 0.0678·45-s + 0.716·47-s − 0.407·51-s + 1.80·53-s − 0.355·55-s + 0.779·57-s − 0.233·59-s − 0.598·61-s − 0.331·65-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.016685605\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.016685605\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 + 0.454T + 5T^{2} \) |
| 11 | \( 1 - 5.79T + 11T^{2} \) |
| 13 | \( 1 - 5.88T + 13T^{2} \) |
| 17 | \( 1 + 2.90T + 17T^{2} \) |
| 19 | \( 1 - 5.88T + 19T^{2} \) |
| 23 | \( 1 - 2.90T + 23T^{2} \) |
| 29 | \( 1 - 3.54T + 29T^{2} \) |
| 31 | \( 1 + 4.33T + 31T^{2} \) |
| 37 | \( 1 - 7.70T + 37T^{2} \) |
| 41 | \( 1 + 9.58T + 41T^{2} \) |
| 43 | \( 1 + 10.7T + 43T^{2} \) |
| 47 | \( 1 - 4.90T + 47T^{2} \) |
| 53 | \( 1 - 13.1T + 53T^{2} \) |
| 59 | \( 1 + 1.79T + 59T^{2} \) |
| 61 | \( 1 + 4.67T + 61T^{2} \) |
| 67 | \( 1 + 7.88T + 67T^{2} \) |
| 71 | \( 1 - 0.909T + 71T^{2} \) |
| 73 | \( 1 + 5.20T + 73T^{2} \) |
| 79 | \( 1 - 2.75T + 79T^{2} \) |
| 83 | \( 1 - 9.97T + 83T^{2} \) |
| 89 | \( 1 - 4.90T + 89T^{2} \) |
| 97 | \( 1 - 5.79T + 97T^{2} \) |
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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
−8.456920413697268526492404860587, −7.61567047035707611949015068874, −6.80093750535903022582386440296, −6.31064040280517815199869036234, −5.40182101631823195871611059999, −4.29826280691441894826595607303, −3.73487988306114784297201057831, −3.10897084028652213437892636799, −1.76413834142388224342774442761, −1.03890672942108314003403536911,
1.03890672942108314003403536911, 1.76413834142388224342774442761, 3.10897084028652213437892636799, 3.73487988306114784297201057831, 4.29826280691441894826595607303, 5.40182101631823195871611059999, 6.31064040280517815199869036234, 6.80093750535903022582386440296, 7.61567047035707611949015068874, 8.456920413697268526492404860587
