| L(s) = 1 | + 2·2-s + 4·3-s + 4·4-s + 8·6-s + 8·8-s − 11·9-s + 60·11-s + 16·12-s + 38·13-s + 16·16-s + 42·17-s − 22·18-s + 52·19-s + 120·22-s − 120·23-s + 32·24-s + 76·26-s − 152·27-s − 234·29-s + 304·31-s + 32·32-s + 240·33-s + 84·34-s − 44·36-s + 106·37-s + 104·38-s + 152·39-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.769·3-s + 1/2·4-s + 0.544·6-s + 0.353·8-s − 0.407·9-s + 1.64·11-s + 0.384·12-s + 0.810·13-s + 1/4·16-s + 0.599·17-s − 0.288·18-s + 0.627·19-s + 1.16·22-s − 1.08·23-s + 0.272·24-s + 0.573·26-s − 1.08·27-s − 1.49·29-s + 1.76·31-s + 0.176·32-s + 1.26·33-s + 0.423·34-s − 0.203·36-s + 0.470·37-s + 0.443·38-s + 0.624·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(5.848386972\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.848386972\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - p T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 - 4 T + p^{3} T^{2} \) |
| 11 | \( 1 - 60 T + p^{3} T^{2} \) |
| 13 | \( 1 - 38 T + p^{3} T^{2} \) |
| 17 | \( 1 - 42 T + p^{3} T^{2} \) |
| 19 | \( 1 - 52 T + p^{3} T^{2} \) |
| 23 | \( 1 + 120 T + p^{3} T^{2} \) |
| 29 | \( 1 + 234 T + p^{3} T^{2} \) |
| 31 | \( 1 - 304 T + p^{3} T^{2} \) |
| 37 | \( 1 - 106 T + p^{3} T^{2} \) |
| 41 | \( 1 - 54 T + p^{3} T^{2} \) |
| 43 | \( 1 - 196 T + p^{3} T^{2} \) |
| 47 | \( 1 - 336 T + p^{3} T^{2} \) |
| 53 | \( 1 + 438 T + p^{3} T^{2} \) |
| 59 | \( 1 - 444 T + p^{3} T^{2} \) |
| 61 | \( 1 + 38 T + p^{3} T^{2} \) |
| 67 | \( 1 - 988 T + p^{3} T^{2} \) |
| 71 | \( 1 + 720 T + p^{3} T^{2} \) |
| 73 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 79 | \( 1 + 808 T + p^{3} T^{2} \) |
| 83 | \( 1 - 612 T + p^{3} T^{2} \) |
| 89 | \( 1 + 1146 T + p^{3} T^{2} \) |
| 97 | \( 1 + 70 T + p^{3} T^{2} \) |
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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
−8.511802367628544639904313165348, −7.88475569723529022532398033213, −7.00259797539800395789828378324, −6.08674046958219281862377101864, −5.65205290417332581378687327792, −4.29278134020823519557349066371, −3.75634044869554433416788795209, −3.01371406009606128820876410646, −1.95432680835436681823995017795, −0.971482314133476007124083405561,
0.971482314133476007124083405561, 1.95432680835436681823995017795, 3.01371406009606128820876410646, 3.75634044869554433416788795209, 4.29278134020823519557349066371, 5.65205290417332581378687327792, 6.08674046958219281862377101864, 7.00259797539800395789828378324, 7.88475569723529022532398033213, 8.511802367628544639904313165348
