| L(s) = 1 | + 2-s + 2·3-s + 4-s + 2·6-s + 7-s + 8-s + 9-s + 2·12-s + 4·13-s + 14-s + 16-s + 6·17-s + 18-s + 2·21-s + 2·24-s − 5·25-s + 4·26-s − 4·27-s + 28-s + 6·29-s + 4·31-s + 32-s + 6·34-s + 36-s − 2·37-s + 8·39-s − 6·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.377·7-s + 0.353·8-s + 1/3·9-s + 0.577·12-s + 1.10·13-s + 0.267·14-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 0.436·21-s + 0.408·24-s − 25-s + 0.784·26-s − 0.769·27-s + 0.188·28-s + 1.11·29-s + 0.718·31-s + 0.176·32-s + 1.02·34-s + 1/6·36-s − 0.328·37-s + 1.28·39-s − 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5054 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.473593818\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.473593818\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 19 | \( 1 \) |
| good | 3 | \( 1 - 2 T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−8.196197307405668368216926690290, −7.72200516209190044787449977815, −6.77298005536370046551425088692, −5.97907088754740589421145166260, −5.31155422432102619146983664079, −4.35906178911561015019581069777, −3.55246093072316681260396868353, −3.09738642858536778661614069445, −2.11458272020859388026266946767, −1.19460132813703026742735308794,
1.19460132813703026742735308794, 2.11458272020859388026266946767, 3.09738642858536778661614069445, 3.55246093072316681260396868353, 4.35906178911561015019581069777, 5.31155422432102619146983664079, 5.97907088754740589421145166260, 6.77298005536370046551425088692, 7.72200516209190044787449977815, 8.196197307405668368216926690290
