| L(s) = 1 | + 2·2-s + 4.69·3-s + 4·4-s + 9.38·6-s − 7·7-s + 8·8-s + 13·9-s + 11·11-s + 18.7·12-s − 23.4·13-s − 14·14-s + 16·16-s + 4.69·17-s + 26·18-s − 32.8·21-s + 22·22-s + 37.5·24-s + 25·25-s − 46.9·26-s + 18.7·27-s − 28·28-s − 60.9·31-s + 32·32-s + 51.5·33-s + 9.38·34-s + 52·36-s − 52·37-s + ⋯ |
| L(s) = 1 | + 2-s + 1.56·3-s + 4-s + 1.56·6-s − 7-s + 8-s + 1.44·9-s + 11-s + 1.56·12-s − 1.80·13-s − 14-s + 16-s + 0.275·17-s + 1.44·18-s − 1.56·21-s + 22-s + 1.56·24-s + 25-s − 1.80·26-s + 0.694·27-s − 28-s − 1.96·31-s + 32-s + 1.56·33-s + 0.275·34-s + 1.44·36-s − 1.40·37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 308 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(4.495764493\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.495764493\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 2T \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
| good | 3 | \( 1 - 4.69T + 9T^{2} \) |
| 5 | \( 1 - 25T^{2} \) |
| 13 | \( 1 + 23.4T + 169T^{2} \) |
| 17 | \( 1 - 4.69T + 289T^{2} \) |
| 19 | \( 1 - 361T^{2} \) |
| 23 | \( 1 - 529T^{2} \) |
| 29 | \( 1 - 841T^{2} \) |
| 31 | \( 1 + 60.9T + 961T^{2} \) |
| 37 | \( 1 + 52T + 1.36e3T^{2} \) |
| 41 | \( 1 + 23.4T + 1.68e3T^{2} \) |
| 43 | \( 1 + 68T + 1.84e3T^{2} \) |
| 47 | \( 1 - 51.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 92T + 2.80e3T^{2} \) |
| 59 | \( 1 + 60.9T + 3.48e3T^{2} \) |
| 61 | \( 1 - 117.T + 3.72e3T^{2} \) |
| 67 | \( 1 - 4.48e3T^{2} \) |
| 71 | \( 1 - 5.04e3T^{2} \) |
| 73 | \( 1 - 4.69T + 5.32e3T^{2} \) |
| 79 | \( 1 - 4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 6.88e3T^{2} \) |
| 89 | \( 1 - 7.92e3T^{2} \) |
| 97 | \( 1 - 9.40e3T^{2} \) |
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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
−11.92605560267518218126018687813, −10.37446397546275165202680436180, −9.571913409874267066568082557064, −8.694902092423101557955270850063, −7.29759118106154229034078679482, −6.88027239309757036110385348537, −5.27986283316227468786620770152, −3.90950989433370673962115930381, −3.13655204782742453777485470286, −2.06481834662068143458164892272,
2.06481834662068143458164892272, 3.13655204782742453777485470286, 3.90950989433370673962115930381, 5.27986283316227468786620770152, 6.88027239309757036110385348537, 7.29759118106154229034078679482, 8.694902092423101557955270850063, 9.571913409874267066568082557064, 10.37446397546275165202680436180, 11.92605560267518218126018687813
