| L(s) = 1 | + 4·2-s + 8·4-s + 7·7-s − 62·11-s + 62·13-s + 28·14-s − 64·16-s + 84·17-s + 100·19-s − 248·22-s − 42·23-s + 248·26-s + 56·28-s + 10·29-s − 48·31-s − 256·32-s + 336·34-s + 246·37-s + 400·38-s + 248·41-s − 68·43-s − 496·44-s − 168·46-s + 324·47-s + 49·49-s + 496·52-s + 258·53-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.377·7-s − 1.69·11-s + 1.32·13-s + 0.534·14-s − 16-s + 1.19·17-s + 1.20·19-s − 2.40·22-s − 0.380·23-s + 1.87·26-s + 0.377·28-s + 0.0640·29-s − 0.278·31-s − 1.41·32-s + 1.69·34-s + 1.09·37-s + 1.70·38-s + 0.944·41-s − 0.241·43-s − 1.69·44-s − 0.538·46-s + 1.00·47-s + 1/7·49-s + 1.32·52-s + 0.668·53-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
| \(L(2)\) |
\(\approx\) |
\(4.893454197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.893454197\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - p T \) |
| good | 2 | \( 1 - p^{2} T + p^{3} T^{2} \) |
| 11 | \( 1 + 62 T + p^{3} T^{2} \) |
| 13 | \( 1 - 62 T + p^{3} T^{2} \) |
| 17 | \( 1 - 84 T + p^{3} T^{2} \) |
| 19 | \( 1 - 100 T + p^{3} T^{2} \) |
| 23 | \( 1 + 42 T + p^{3} T^{2} \) |
| 29 | \( 1 - 10 T + p^{3} T^{2} \) |
| 31 | \( 1 + 48 T + p^{3} T^{2} \) |
| 37 | \( 1 - 246 T + p^{3} T^{2} \) |
| 41 | \( 1 - 248 T + p^{3} T^{2} \) |
| 43 | \( 1 + 68 T + p^{3} T^{2} \) |
| 47 | \( 1 - 324 T + p^{3} T^{2} \) |
| 53 | \( 1 - 258 T + p^{3} T^{2} \) |
| 59 | \( 1 + 120 T + p^{3} T^{2} \) |
| 61 | \( 1 - 622 T + p^{3} T^{2} \) |
| 67 | \( 1 + 904 T + p^{3} T^{2} \) |
| 71 | \( 1 - 678 T + p^{3} T^{2} \) |
| 73 | \( 1 - 642 T + p^{3} T^{2} \) |
| 79 | \( 1 - 740 T + p^{3} T^{2} \) |
| 83 | \( 1 - 468 T + p^{3} T^{2} \) |
| 89 | \( 1 + 200 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1266 T + p^{3} T^{2} \) |
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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
−9.020865447298300437533336465256, −7.988810398149611745325608056415, −7.48593024241648033519195046686, −6.22212283953202391821269521630, −5.54508030259633477513086612851, −5.07739690829328989853828947239, −3.98439792112703463023198376304, −3.22641944548716133059802995764, −2.35599200569381471623615688378, −0.875832328420652040712321749228,
0.875832328420652040712321749228, 2.35599200569381471623615688378, 3.22641944548716133059802995764, 3.98439792112703463023198376304, 5.07739690829328989853828947239, 5.54508030259633477513086612851, 6.22212283953202391821269521630, 7.48593024241648033519195046686, 7.988810398149611745325608056415, 9.020865447298300437533336465256
