| L(s) = 1 | + 4.70·2-s + 14.1·4-s − 7·7-s + 28.7·8-s − 24.5·11-s + 35.0·13-s − 32.9·14-s + 22.1·16-s − 18.4·17-s − 67.4·19-s − 115.·22-s − 145.·23-s + 164.·26-s − 98.7·28-s − 214.·29-s − 88.6·31-s − 125.·32-s − 86.5·34-s − 162.·37-s − 316.·38-s + 337.·41-s − 122.·43-s − 346.·44-s − 684.·46-s + 354.·47-s + 49·49-s + 493.·52-s + ⋯ |
| L(s) = 1 | + 1.66·2-s + 1.76·4-s − 0.377·7-s + 1.26·8-s − 0.674·11-s + 0.747·13-s − 0.628·14-s + 0.345·16-s − 0.262·17-s − 0.813·19-s − 1.12·22-s − 1.32·23-s + 1.24·26-s − 0.666·28-s − 1.37·29-s − 0.513·31-s − 0.694·32-s − 0.436·34-s − 0.720·37-s − 1.35·38-s + 1.28·41-s − 0.433·43-s − 1.18·44-s − 2.19·46-s + 1.09·47-s + 0.142·49-s + 1.31·52-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 4.70T + 8T^{2} \) |
| 11 | \( 1 + 24.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 35.0T + 2.19e3T^{2} \) |
| 17 | \( 1 + 18.4T + 4.91e3T^{2} \) |
| 19 | \( 1 + 67.4T + 6.85e3T^{2} \) |
| 23 | \( 1 + 145.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 214.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 88.6T + 2.97e4T^{2} \) |
| 37 | \( 1 + 162.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 337.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 122.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 354.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 676.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 501.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 708.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 907.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 430.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 41.3T + 3.89e5T^{2} \) |
| 79 | \( 1 - 890.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.05e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.47e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 555.T + 9.12e5T^{2} \) |
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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
−8.575373559479012227022536871869, −7.57878062627741873562892991580, −6.75132231929841740465151123716, −5.88656040652363805603672663416, −5.48125778991697088863055260271, −4.26124502480101793559116704086, −3.80202910354351671165325917116, −2.73969600391756561547826051309, −1.86520601809035991412245784928, 0,
1.86520601809035991412245784928, 2.73969600391756561547826051309, 3.80202910354351671165325917116, 4.26124502480101793559116704086, 5.48125778991697088863055260271, 5.88656040652363805603672663416, 6.75132231929841740465151123716, 7.57878062627741873562892991580, 8.575373559479012227022536871869
