| L(s) = 1 | − 2·2-s − 2.43·3-s + 4·4-s + 4.87·6-s + 3.42·7-s − 8·8-s − 21.0·9-s − 53.2·11-s − 9.74·12-s − 28.9·13-s − 6.84·14-s + 16·16-s − 116.·17-s + 42.1·18-s − 57.7·19-s − 8.33·21-s + 106.·22-s − 23·23-s + 19.4·24-s + 57.8·26-s + 117.·27-s + 13.6·28-s − 54.5·29-s − 263.·31-s − 32·32-s + 129.·33-s + 232.·34-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.468·3-s + 0.5·4-s + 0.331·6-s + 0.184·7-s − 0.353·8-s − 0.780·9-s − 1.46·11-s − 0.234·12-s − 0.617·13-s − 0.130·14-s + 0.250·16-s − 1.66·17-s + 0.551·18-s − 0.697·19-s − 0.0866·21-s + 1.03·22-s − 0.208·23-s + 0.165·24-s + 0.436·26-s + 0.834·27-s + 0.0924·28-s − 0.349·29-s − 1.52·31-s − 0.176·32-s + 0.684·33-s + 1.17·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.2362771051\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2362771051\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 5 | \( 1 \) |
| 23 | \( 1 + 23T \) |
| good | 3 | \( 1 + 2.43T + 27T^{2} \) |
| 7 | \( 1 - 3.42T + 343T^{2} \) |
| 11 | \( 1 + 53.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 28.9T + 2.19e3T^{2} \) |
| 17 | \( 1 + 116.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 57.7T + 6.85e3T^{2} \) |
| 29 | \( 1 + 54.5T + 2.43e4T^{2} \) |
| 31 | \( 1 + 263.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 142.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 316.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 235.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 504.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 380.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 653.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 507.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 740.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 302.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 265.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.18e3T + 4.93e5T^{2} \) |
| 83 | \( 1 + 489.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 484.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.19e3T + 9.12e5T^{2} \) |
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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
−9.260872777399484561896111017627, −8.708652690905144018469346281318, −7.79387647386892645248194141672, −7.10089383030897924871935234718, −6.03979597387119851566872438714, −5.34033034945285156422549769687, −4.32468075239228882764657638127, −2.77634941216616372440306156574, −2.06106805413908766697694842196, −0.26488055787276114857115778769,
0.26488055787276114857115778769, 2.06106805413908766697694842196, 2.77634941216616372440306156574, 4.32468075239228882764657638127, 5.34033034945285156422549769687, 6.03979597387119851566872438714, 7.10089383030897924871935234718, 7.79387647386892645248194141672, 8.708652690905144018469346281318, 9.260872777399484561896111017627
