| L(s) = 1 | + 2-s + 4-s + 4·7-s + 8-s + 5·11-s − 3·13-s + 4·14-s + 16-s + 17-s − 6·19-s + 5·22-s + 23-s − 3·26-s + 4·28-s + 9·29-s − 5·31-s + 32-s + 34-s − 2·37-s − 6·38-s + 2·41-s + 43-s + 5·44-s + 46-s − 13·47-s + 9·49-s − 3·52-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.51·7-s + 0.353·8-s + 1.50·11-s − 0.832·13-s + 1.06·14-s + 1/4·16-s + 0.242·17-s − 1.37·19-s + 1.06·22-s + 0.208·23-s − 0.588·26-s + 0.755·28-s + 1.67·29-s − 0.898·31-s + 0.176·32-s + 0.171·34-s − 0.328·37-s − 0.973·38-s + 0.312·41-s + 0.152·43-s + 0.753·44-s + 0.147·46-s − 1.89·47-s + 9/7·49-s − 0.416·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.202551975\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.202551975\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 + 3 T + p T^{2} \) |
| 17 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 - 9 T + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 + 13 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 9 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 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
−9.629040703030232723473893249028, −8.637555110172477268184123413302, −8.011167219466254498843807764176, −6.98180323166365251692687741203, −6.34365861958936828868449634874, −5.16828547204877680886698324301, −4.56826252399144971888742971934, −3.75736110095399089729442807020, −2.34643797963527675698080469472, −1.38203316215783094853604826070,
1.38203316215783094853604826070, 2.34643797963527675698080469472, 3.75736110095399089729442807020, 4.56826252399144971888742971934, 5.16828547204877680886698324301, 6.34365861958936828868449634874, 6.98180323166365251692687741203, 8.011167219466254498843807764176, 8.637555110172477268184123413302, 9.629040703030232723473893249028
