| L(s) = 1 | − 2-s + 2·3-s + 4-s − 2·6-s − 8-s + 9-s + 2·12-s − 2·13-s + 16-s + 3·17-s − 18-s − 8·19-s − 9·23-s − 2·24-s + 2·26-s − 4·27-s − 6·29-s − 5·31-s − 32-s − 3·34-s + 36-s + 8·37-s + 8·38-s − 4·39-s + 3·41-s − 10·43-s + 9·46-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.15·3-s + 1/2·4-s − 0.816·6-s − 0.353·8-s + 1/3·9-s + 0.577·12-s − 0.554·13-s + 1/4·16-s + 0.727·17-s − 0.235·18-s − 1.83·19-s − 1.87·23-s − 0.408·24-s + 0.392·26-s − 0.769·27-s − 1.11·29-s − 0.898·31-s − 0.176·32-s − 0.514·34-s + 1/6·36-s + 1.31·37-s + 1.29·38-s − 0.640·39-s + 0.468·41-s − 1.52·43-s + 1.32·46-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 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 - 4 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + 9 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 5 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−8.489545705885405952158883826568, −7.962950553847638552578296521149, −7.41352656211076017714151559495, −6.35800879793075590395852720552, −5.61345201989018898426402382777, −4.26178136637081825613249263760, −3.52001225636062142108317463900, −2.41651031757160203086446434532, −1.84953673769386929931681068545, 0,
1.84953673769386929931681068545, 2.41651031757160203086446434532, 3.52001225636062142108317463900, 4.26178136637081825613249263760, 5.61345201989018898426402382777, 6.35800879793075590395852720552, 7.41352656211076017714151559495, 7.962950553847638552578296521149, 8.489545705885405952158883826568
