| L(s) = 1 | + 3·5-s − 11-s − 7·13-s − 6·17-s + 19-s + 4·25-s − 3·29-s + 2·31-s − 5·37-s − 12·41-s + 8·43-s + 3·47-s − 12·53-s − 3·55-s − 3·59-s − 10·61-s − 21·65-s − 13·67-s − 12·71-s − 11·73-s − 8·79-s + 6·83-s − 18·85-s + 6·89-s + 3·95-s − 8·97-s + 101-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.301·11-s − 1.94·13-s − 1.45·17-s + 0.229·19-s + 4/5·25-s − 0.557·29-s + 0.359·31-s − 0.821·37-s − 1.87·41-s + 1.21·43-s + 0.437·47-s − 1.64·53-s − 0.404·55-s − 0.390·59-s − 1.28·61-s − 2.60·65-s − 1.58·67-s − 1.42·71-s − 1.28·73-s − 0.900·79-s + 0.658·83-s − 1.95·85-s + 0.635·89-s + 0.307·95-s − 0.812·97-s + 0.0995·101-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 310464 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1284797236\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1284797236\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 7 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 3 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.73558274291452, −12.19273081195300, −11.85481996773115, −11.28463144075402, −10.61738953192370, −10.35508801540592, −9.953976306602888, −9.420941707284056, −9.074352778405515, −8.740691974867698, −7.960184454548053, −7.404217219175579, −7.144672204156986, −6.506236966975187, −6.095220130272918, −5.558465856329199, −5.098899162746109, −4.555981602869740, −4.344768112914520, −3.164205029911038, −2.938389866450716, −2.174615205720004, −1.943110996519007, −1.318570440196864, −0.08583969860864410,
0.08583969860864410, 1.318570440196864, 1.943110996519007, 2.174615205720004, 2.938389866450716, 3.164205029911038, 4.344768112914520, 4.555981602869740, 5.098899162746109, 5.558465856329199, 6.095220130272918, 6.506236966975187, 7.144672204156986, 7.404217219175579, 7.960184454548053, 8.740691974867698, 9.074352778405515, 9.420941707284056, 9.953976306602888, 10.35508801540592, 10.61738953192370, 11.28463144075402, 11.85481996773115, 12.19273081195300, 12.73558274291452
