| L(s) = 1 | − 0.993·3-s + 3.24·5-s − 4.74·7-s − 2.01·9-s − 5.30·11-s − 13-s − 3.22·15-s − 5.32·17-s − 5.67·19-s + 4.71·21-s − 3.62·23-s + 5.53·25-s + 4.98·27-s + 29-s + 0.521·31-s + 5.27·33-s − 15.4·35-s − 6.10·37-s + 0.993·39-s − 5.95·41-s + 11.9·43-s − 6.52·45-s + 1.32·47-s + 15.5·49-s + 5.29·51-s − 4.90·53-s − 17.2·55-s + ⋯ |
| L(s) = 1 | − 0.573·3-s + 1.45·5-s − 1.79·7-s − 0.670·9-s − 1.59·11-s − 0.277·13-s − 0.832·15-s − 1.29·17-s − 1.30·19-s + 1.02·21-s − 0.755·23-s + 1.10·25-s + 0.958·27-s + 0.185·29-s + 0.0937·31-s + 0.917·33-s − 2.60·35-s − 1.00·37-s + 0.159·39-s − 0.930·41-s + 1.82·43-s − 0.973·45-s + 0.192·47-s + 2.21·49-s + 0.741·51-s − 0.673·53-s − 2.32·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4507547478\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4507547478\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 13 | \( 1 + T \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + 0.993T + 3T^{2} \) |
| 5 | \( 1 - 3.24T + 5T^{2} \) |
| 7 | \( 1 + 4.74T + 7T^{2} \) |
| 11 | \( 1 + 5.30T + 11T^{2} \) |
| 17 | \( 1 + 5.32T + 17T^{2} \) |
| 19 | \( 1 + 5.67T + 19T^{2} \) |
| 23 | \( 1 + 3.62T + 23T^{2} \) |
| 31 | \( 1 - 0.521T + 31T^{2} \) |
| 37 | \( 1 + 6.10T + 37T^{2} \) |
| 41 | \( 1 + 5.95T + 41T^{2} \) |
| 43 | \( 1 - 11.9T + 43T^{2} \) |
| 47 | \( 1 - 1.32T + 47T^{2} \) |
| 53 | \( 1 + 4.90T + 53T^{2} \) |
| 59 | \( 1 + 13.3T + 59T^{2} \) |
| 61 | \( 1 - 12.3T + 61T^{2} \) |
| 67 | \( 1 - 1.21T + 67T^{2} \) |
| 71 | \( 1 - 14.4T + 71T^{2} \) |
| 73 | \( 1 + 0.437T + 73T^{2} \) |
| 79 | \( 1 - 11.8T + 79T^{2} \) |
| 83 | \( 1 + 3.97T + 83T^{2} \) |
| 89 | \( 1 - 0.468T + 89T^{2} \) |
| 97 | \( 1 + 8.41T + 97T^{2} \) |
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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
−8.191299902879066290981677151912, −7.05812768443349882984007555429, −6.36901210733739604098466174607, −6.08847357750782275281691798708, −5.41348041427655588762596840343, −4.68722401646657403968383053759, −3.48335922258403511451028806791, −2.46463459826727916105822204337, −2.26790298696080672658101129478, −0.32734449028126003799998692174,
0.32734449028126003799998692174, 2.26790298696080672658101129478, 2.46463459826727916105822204337, 3.48335922258403511451028806791, 4.68722401646657403968383053759, 5.41348041427655588762596840343, 6.08847357750782275281691798708, 6.36901210733739604098466174607, 7.05812768443349882984007555429, 8.191299902879066290981677151912
