| L(s) = 1 | − 4·5-s + 13-s + 2·17-s + 8·19-s + 4·23-s + 11·25-s − 6·29-s + 4·31-s + 6·37-s − 12·41-s + 4·43-s − 6·47-s − 7·49-s + 2·53-s − 14·59-s − 10·61-s − 4·65-s + 4·67-s + 2·71-s + 2·73-s − 8·79-s − 14·83-s − 8·85-s − 32·95-s − 10·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 1.78·5-s + 0.277·13-s + 0.485·17-s + 1.83·19-s + 0.834·23-s + 11/5·25-s − 1.11·29-s + 0.718·31-s + 0.986·37-s − 1.87·41-s + 0.609·43-s − 0.875·47-s − 49-s + 0.274·53-s − 1.82·59-s − 1.28·61-s − 0.496·65-s + 0.488·67-s + 0.237·71-s + 0.234·73-s − 0.900·79-s − 1.53·83-s − 0.867·85-s − 3.28·95-s − 1.01·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 226512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9912755771\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9912755771\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 5 | \( 1 + 4 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 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
−12.87248175718752, −12.35392748752668, −11.96547420062572, −11.51600674010855, −11.22445578852802, −10.86297761708782, −10.10628299569212, −9.669427329467274, −9.172971705417046, −8.624952864991245, −8.051472943695720, −7.825822787399483, −7.261480244177876, −7.008411257832921, −6.275386854151131, −5.693368482895716, −4.970151918501308, −4.773389546010186, −4.048017045327733, −3.522088989842828, −3.134431457790025, −2.760395909529837, −1.516143312295906, −1.150082288571583, −0.3098100237753480,
0.3098100237753480, 1.150082288571583, 1.516143312295906, 2.760395909529837, 3.134431457790025, 3.522088989842828, 4.048017045327733, 4.773389546010186, 4.970151918501308, 5.693368482895716, 6.275386854151131, 7.008411257832921, 7.261480244177876, 7.825822787399483, 8.051472943695720, 8.624952864991245, 9.172971705417046, 9.669427329467274, 10.10628299569212, 10.86297761708782, 11.22445578852802, 11.51600674010855, 11.96547420062572, 12.35392748752668, 12.87248175718752
