| L(s) = 1 | + 14·5-s + 7·7-s − 28·11-s + 18·13-s − 74·17-s − 80·19-s − 112·23-s + 71·25-s − 190·29-s − 72·31-s + 98·35-s − 346·37-s − 162·41-s + 412·43-s + 24·47-s + 49·49-s − 318·53-s − 392·55-s − 200·59-s − 198·61-s + 252·65-s + 716·67-s + 392·71-s + 538·73-s − 196·77-s − 240·79-s − 1.07e3·83-s + ⋯ |
| L(s) = 1 | + 1.25·5-s + 0.377·7-s − 0.767·11-s + 0.384·13-s − 1.05·17-s − 0.965·19-s − 1.01·23-s + 0.567·25-s − 1.21·29-s − 0.417·31-s + 0.473·35-s − 1.53·37-s − 0.617·41-s + 1.46·43-s + 0.0744·47-s + 1/7·49-s − 0.824·53-s − 0.961·55-s − 0.441·59-s − 0.415·61-s + 0.480·65-s + 1.30·67-s + 0.655·71-s + 0.862·73-s − 0.290·77-s − 0.341·79-s − 1.41·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 - p T \) |
| good | 5 | \( 1 - 14 T + p^{3} T^{2} \) |
| 11 | \( 1 + 28 T + p^{3} T^{2} \) |
| 13 | \( 1 - 18 T + p^{3} T^{2} \) |
| 17 | \( 1 + 74 T + p^{3} T^{2} \) |
| 19 | \( 1 + 80 T + p^{3} T^{2} \) |
| 23 | \( 1 + 112 T + p^{3} T^{2} \) |
| 29 | \( 1 + 190 T + p^{3} T^{2} \) |
| 31 | \( 1 + 72 T + p^{3} T^{2} \) |
| 37 | \( 1 + 346 T + p^{3} T^{2} \) |
| 41 | \( 1 + 162 T + p^{3} T^{2} \) |
| 43 | \( 1 - 412 T + p^{3} T^{2} \) |
| 47 | \( 1 - 24 T + p^{3} T^{2} \) |
| 53 | \( 1 + 6 p T + p^{3} T^{2} \) |
| 59 | \( 1 + 200 T + p^{3} T^{2} \) |
| 61 | \( 1 + 198 T + p^{3} T^{2} \) |
| 67 | \( 1 - 716 T + p^{3} T^{2} \) |
| 71 | \( 1 - 392 T + p^{3} T^{2} \) |
| 73 | \( 1 - 538 T + p^{3} T^{2} \) |
| 79 | \( 1 + 240 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1072 T + p^{3} T^{2} \) |
| 89 | \( 1 + 810 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1354 T + p^{3} T^{2} \) |
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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
−9.153016579211602275091005071088, −8.483153674109618061196008026156, −7.49332803818190022350010272880, −6.44393080814356828591823515696, −5.76045298658144087896757087716, −4.92539414238910028884302258399, −3.80780751621761782879752293974, −2.33758019283373778979251015658, −1.76041416984475460044699449942, 0,
1.76041416984475460044699449942, 2.33758019283373778979251015658, 3.80780751621761782879752293974, 4.92539414238910028884302258399, 5.76045298658144087896757087716, 6.44393080814356828591823515696, 7.49332803818190022350010272880, 8.483153674109618061196008026156, 9.153016579211602275091005071088
