| L(s) = 1 | + 1.30·2-s + 3-s − 0.288·4-s + 5-s + 1.30·6-s + 4.16·7-s − 2.99·8-s + 9-s + 1.30·10-s − 0.738·11-s − 0.288·12-s + 13-s + 5.45·14-s + 15-s − 3.33·16-s + 6.77·17-s + 1.30·18-s + 1.64·19-s − 0.288·20-s + 4.16·21-s − 0.966·22-s − 1.96·23-s − 2.99·24-s + 25-s + 1.30·26-s + 27-s − 1.20·28-s + ⋯ |
| L(s) = 1 | + 0.924·2-s + 0.577·3-s − 0.144·4-s + 0.447·5-s + 0.534·6-s + 1.57·7-s − 1.05·8-s + 0.333·9-s + 0.413·10-s − 0.222·11-s − 0.0833·12-s + 0.277·13-s + 1.45·14-s + 0.258·15-s − 0.834·16-s + 1.64·17-s + 0.308·18-s + 0.377·19-s − 0.0645·20-s + 0.909·21-s − 0.206·22-s − 0.410·23-s − 0.611·24-s + 0.200·25-s + 0.256·26-s + 0.192·27-s − 0.227·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6045 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.938250567\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.938250567\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| 31 | \( 1 + T \) |
| good | 2 | \( 1 - 1.30T + 2T^{2} \) |
| 7 | \( 1 - 4.16T + 7T^{2} \) |
| 11 | \( 1 + 0.738T + 11T^{2} \) |
| 17 | \( 1 - 6.77T + 17T^{2} \) |
| 19 | \( 1 - 1.64T + 19T^{2} \) |
| 23 | \( 1 + 1.96T + 23T^{2} \) |
| 29 | \( 1 - 2.33T + 29T^{2} \) |
| 37 | \( 1 + 1.52T + 37T^{2} \) |
| 41 | \( 1 + 3.86T + 41T^{2} \) |
| 43 | \( 1 + 6.40T + 43T^{2} \) |
| 47 | \( 1 - 13.0T + 47T^{2} \) |
| 53 | \( 1 + 4.93T + 53T^{2} \) |
| 59 | \( 1 + 9.82T + 59T^{2} \) |
| 61 | \( 1 - 8.71T + 61T^{2} \) |
| 67 | \( 1 - 8.45T + 67T^{2} \) |
| 71 | \( 1 + 6.38T + 71T^{2} \) |
| 73 | \( 1 + 1.29T + 73T^{2} \) |
| 79 | \( 1 - 3.57T + 79T^{2} \) |
| 83 | \( 1 - 4.85T + 83T^{2} \) |
| 89 | \( 1 - 1.96T + 89T^{2} \) |
| 97 | \( 1 - 1.03T + 97T^{2} \) |
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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
−8.055264094690090913747431875628, −7.53710936216381860507356350362, −6.49928523131778790824217450714, −5.55006086869470842949412542335, −5.23563721922505974965867190978, −4.48735083478993304828236547507, −3.70212907739791745632403255995, −2.97101372833658453760025555683, −1.99227779755239881617809929601, −1.07442449289528455308824186825,
1.07442449289528455308824186825, 1.99227779755239881617809929601, 2.97101372833658453760025555683, 3.70212907739791745632403255995, 4.48735083478993304828236547507, 5.23563721922505974965867190978, 5.55006086869470842949412542335, 6.49928523131778790824217450714, 7.53710936216381860507356350362, 8.055264094690090913747431875628
