| L(s) = 1 | − 5.56·2-s + 23.0·4-s − 7·7-s − 83.5·8-s − 23.2·11-s − 46.5·13-s + 38.9·14-s + 281.·16-s − 76.0·17-s − 114.·19-s + 129.·22-s − 113.·23-s + 259.·26-s − 161.·28-s − 120.·29-s − 182.·31-s − 897.·32-s + 423.·34-s − 322.·37-s + 638.·38-s + 93.0·41-s − 452.·43-s − 535.·44-s + 634.·46-s − 402.·47-s + 49·49-s − 1.07e3·52-s + ⋯ |
| L(s) = 1 | − 1.96·2-s + 2.87·4-s − 0.377·7-s − 3.69·8-s − 0.638·11-s − 0.992·13-s + 0.744·14-s + 4.39·16-s − 1.08·17-s − 1.38·19-s + 1.25·22-s − 1.03·23-s + 1.95·26-s − 1.08·28-s − 0.771·29-s − 1.05·31-s − 4.95·32-s + 2.13·34-s − 1.43·37-s + 2.72·38-s + 0.354·41-s − 1.60·43-s − 1.83·44-s + 2.03·46-s − 1.24·47-s + 0.142·49-s − 2.85·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.01308078744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.01308078744\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| good | 2 | \( 1 + 5.56T + 8T^{2} \) |
| 11 | \( 1 + 23.2T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 76.0T + 4.91e3T^{2} \) |
| 19 | \( 1 + 114.T + 6.85e3T^{2} \) |
| 23 | \( 1 + 113.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 120.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 182.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 322.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 93.0T + 6.89e4T^{2} \) |
| 43 | \( 1 + 452.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 402.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 495.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 496.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 265.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 594.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 510.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 470.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 487.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.25e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.56e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 21.5T + 9.12e5T^{2} \) |
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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
−9.084844937191258269173868906612, −8.360582758666421432931526058702, −7.69811596251420827213058696075, −6.86537418245821953652081470031, −6.33071017069943119181921407061, −5.19611775594406883579686158676, −3.60890562131305218769227331663, −2.36126301083665385320515970421, −1.87206441169170242931347454932, −0.06865898291221914768908848617,
0.06865898291221914768908848617, 1.87206441169170242931347454932, 2.36126301083665385320515970421, 3.60890562131305218769227331663, 5.19611775594406883579686158676, 6.33071017069943119181921407061, 6.86537418245821953652081470031, 7.69811596251420827213058696075, 8.360582758666421432931526058702, 9.084844937191258269173868906612
