| L(s) = 1 | − 2.62·2-s + 1.33·3-s + 4.88·4-s + 5-s − 3.51·6-s + 2.62·7-s − 7.56·8-s − 1.20·9-s − 2.62·10-s + 6.54·12-s + 6.72·13-s − 6.88·14-s + 1.33·15-s + 10.0·16-s − 1.78·17-s + 3.16·18-s + 1.48·19-s + 4.88·20-s + 3.51·21-s + 5.20·23-s − 10.1·24-s + 25-s − 17.6·26-s − 5.63·27-s + 12.8·28-s − 1.17·29-s − 3.51·30-s + ⋯ |
| L(s) = 1 | − 1.85·2-s + 0.773·3-s + 2.44·4-s + 0.447·5-s − 1.43·6-s + 0.991·7-s − 2.67·8-s − 0.401·9-s − 0.829·10-s + 1.88·12-s + 1.86·13-s − 1.83·14-s + 0.345·15-s + 2.52·16-s − 0.432·17-s + 0.745·18-s + 0.339·19-s + 1.09·20-s + 0.767·21-s + 1.08·23-s − 2.06·24-s + 0.200·25-s − 3.46·26-s − 1.08·27-s + 2.42·28-s − 0.218·29-s − 0.641·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.030140576\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.030140576\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| good | 2 | \( 1 + 2.62T + 2T^{2} \) |
| 3 | \( 1 - 1.33T + 3T^{2} \) |
| 7 | \( 1 - 2.62T + 7T^{2} \) |
| 13 | \( 1 - 6.72T + 13T^{2} \) |
| 17 | \( 1 + 1.78T + 17T^{2} \) |
| 19 | \( 1 - 1.48T + 19T^{2} \) |
| 23 | \( 1 - 5.20T + 23T^{2} \) |
| 29 | \( 1 + 1.17T + 29T^{2} \) |
| 31 | \( 1 + 4.56T + 31T^{2} \) |
| 37 | \( 1 + 0.679T + 37T^{2} \) |
| 41 | \( 1 + 5.49T + 41T^{2} \) |
| 43 | \( 1 + 3.80T + 43T^{2} \) |
| 47 | \( 1 - 6.66T + 47T^{2} \) |
| 53 | \( 1 - 14.2T + 53T^{2} \) |
| 59 | \( 1 + 3.88T + 59T^{2} \) |
| 61 | \( 1 + 3.21T + 61T^{2} \) |
| 67 | \( 1 - 4.66T + 67T^{2} \) |
| 71 | \( 1 - 3.88T + 71T^{2} \) |
| 73 | \( 1 - 3.56T + 73T^{2} \) |
| 79 | \( 1 - 15.1T + 79T^{2} \) |
| 83 | \( 1 + 7.33T + 83T^{2} \) |
| 89 | \( 1 - 3.44T + 89T^{2} \) |
| 97 | \( 1 - 1.47T + 97T^{2} \) |
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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
−10.63443407143624285770296850225, −9.434492035052023876494798513923, −8.764055223501386279933793181115, −8.419820344753635553492105010448, −7.53104660546720145014146238481, −6.53366734463513775390624596425, −5.46390942896371785159299389916, −3.49967526285999022435065296196, −2.24498261502305108827566285417, −1.25138706605810035988356508603,
1.25138706605810035988356508603, 2.24498261502305108827566285417, 3.49967526285999022435065296196, 5.46390942896371785159299389916, 6.53366734463513775390624596425, 7.53104660546720145014146238481, 8.419820344753635553492105010448, 8.764055223501386279933793181115, 9.434492035052023876494798513923, 10.63443407143624285770296850225
