| L(s) = 1 | + 3·5-s + 2·7-s − 6·17-s − 4·19-s − 23-s + 4·25-s − 8·29-s − 7·31-s + 6·35-s − 37-s + 4·41-s − 6·43-s + 8·47-s − 3·49-s − 2·53-s + 59-s − 4·61-s − 5·67-s − 3·71-s − 16·73-s − 2·79-s − 2·83-s − 18·85-s − 15·89-s − 12·95-s − 7·97-s − 10·101-s + ⋯ |
| L(s) = 1 | + 1.34·5-s + 0.755·7-s − 1.45·17-s − 0.917·19-s − 0.208·23-s + 4/5·25-s − 1.48·29-s − 1.25·31-s + 1.01·35-s − 0.164·37-s + 0.624·41-s − 0.914·43-s + 1.16·47-s − 3/7·49-s − 0.274·53-s + 0.130·59-s − 0.512·61-s − 0.610·67-s − 0.356·71-s − 1.87·73-s − 0.225·79-s − 0.219·83-s − 1.95·85-s − 1.58·89-s − 1.23·95-s − 0.710·97-s − 0.995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8712 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 + 5 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 + 15 T + p T^{2} \) |
| 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
−7.29720117592186502430787073088, −6.74204977853130932719748164169, −5.85833023693336809454573391055, −5.56676285148888490952927599490, −4.60219576415652061161433212140, −4.06031293128076655255788499594, −2.85382636005766047067802737212, −1.93406343891250018924092797371, −1.68010100412897754670208949086, 0,
1.68010100412897754670208949086, 1.93406343891250018924092797371, 2.85382636005766047067802737212, 4.06031293128076655255788499594, 4.60219576415652061161433212140, 5.56676285148888490952927599490, 5.85833023693336809454573391055, 6.74204977853130932719748164169, 7.29720117592186502430787073088
