| L(s) = 1 | − 5·5-s + 4·11-s + 54·13-s − 114·17-s − 44·19-s + 96·23-s + 25·25-s − 134·29-s + 272·31-s − 98·37-s + 6·41-s − 12·43-s − 200·47-s − 343·49-s − 654·53-s − 20·55-s + 36·59-s − 442·61-s − 270·65-s + 188·67-s − 632·71-s − 390·73-s − 688·79-s + 1.18e3·83-s + 570·85-s + 694·89-s + 220·95-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.109·11-s + 1.15·13-s − 1.62·17-s − 0.531·19-s + 0.870·23-s + 1/5·25-s − 0.858·29-s + 1.57·31-s − 0.435·37-s + 0.0228·41-s − 0.0425·43-s − 0.620·47-s − 49-s − 1.69·53-s − 0.0490·55-s + 0.0794·59-s − 0.927·61-s − 0.515·65-s + 0.342·67-s − 1.05·71-s − 0.625·73-s − 0.979·79-s + 1.57·83-s + 0.727·85-s + 0.826·89-s + 0.237·95-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + p T \) |
| good | 7 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 - 4 T + p^{3} T^{2} \) |
| 13 | \( 1 - 54 T + p^{3} T^{2} \) |
| 17 | \( 1 + 114 T + p^{3} T^{2} \) |
| 19 | \( 1 + 44 T + p^{3} T^{2} \) |
| 23 | \( 1 - 96 T + p^{3} T^{2} \) |
| 29 | \( 1 + 134 T + p^{3} T^{2} \) |
| 31 | \( 1 - 272 T + p^{3} T^{2} \) |
| 37 | \( 1 + 98 T + p^{3} T^{2} \) |
| 41 | \( 1 - 6 T + p^{3} T^{2} \) |
| 43 | \( 1 + 12 T + p^{3} T^{2} \) |
| 47 | \( 1 + 200 T + p^{3} T^{2} \) |
| 53 | \( 1 + 654 T + p^{3} T^{2} \) |
| 59 | \( 1 - 36 T + p^{3} T^{2} \) |
| 61 | \( 1 + 442 T + p^{3} T^{2} \) |
| 67 | \( 1 - 188 T + p^{3} T^{2} \) |
| 71 | \( 1 + 632 T + p^{3} T^{2} \) |
| 73 | \( 1 + 390 T + p^{3} T^{2} \) |
| 79 | \( 1 + 688 T + p^{3} T^{2} \) |
| 83 | \( 1 - 1188 T + p^{3} T^{2} \) |
| 89 | \( 1 - 694 T + p^{3} T^{2} \) |
| 97 | \( 1 + 1726 T + p^{3} T^{2} \) |
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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
−9.441592853999042607847323646935, −8.691636308798170730205431417433, −7.995268274334997955762032999023, −6.80148373918685823544763409531, −6.22288113544855041840252182719, −4.87083806250021007745933914505, −4.04222466359204504078249596149, −2.92112734152836684564415836551, −1.50886266292918633478968469462, 0,
1.50886266292918633478968469462, 2.92112734152836684564415836551, 4.04222466359204504078249596149, 4.87083806250021007745933914505, 6.22288113544855041840252182719, 6.80148373918685823544763409531, 7.995268274334997955762032999023, 8.691636308798170730205431417433, 9.441592853999042607847323646935
