| L(s) = 1 | − 2-s + 4-s − 8-s − 3·9-s − 2·11-s + 16-s + 7·17-s + 3·18-s + 2·22-s − 3·23-s + 6·29-s − 7·31-s − 32-s − 7·34-s − 3·36-s + 4·37-s − 7·41-s + 8·43-s − 2·44-s + 3·46-s + 7·47-s − 4·53-s − 6·58-s − 14·59-s − 14·61-s + 7·62-s + 64-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.353·8-s − 9-s − 0.603·11-s + 1/4·16-s + 1.69·17-s + 0.707·18-s + 0.426·22-s − 0.625·23-s + 1.11·29-s − 1.25·31-s − 0.176·32-s − 1.20·34-s − 1/2·36-s + 0.657·37-s − 1.09·41-s + 1.21·43-s − 0.301·44-s + 0.442·46-s + 1.02·47-s − 0.549·53-s − 0.787·58-s − 1.82·59-s − 1.79·61-s + 0.889·62-s + 1/8·64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 7 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 + 11 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 97 | \( 1 - 7 T + p T^{2} \) |
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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
−8.602179364766552738898169195328, −7.75501102296877877506456840354, −7.40211456817844464033003393980, −6.03414848905499955917138325122, −5.75331892346643392596355450040, −4.64438314057580767115164598267, −3.33155806464358311256843611794, −2.69599018315806890823449785766, −1.40629630566667339221134627080, 0,
1.40629630566667339221134627080, 2.69599018315806890823449785766, 3.33155806464358311256843611794, 4.64438314057580767115164598267, 5.75331892346643392596355450040, 6.03414848905499955917138325122, 7.40211456817844464033003393980, 7.75501102296877877506456840354, 8.602179364766552738898169195328
