| L(s) = 1 | + 3-s + 4·5-s + 9-s + 2·11-s + 2·13-s + 4·15-s − 4·19-s + 6·23-s + 11·25-s + 27-s − 10·29-s − 8·31-s + 2·33-s + 10·37-s + 2·39-s + 4·41-s + 8·43-s + 4·45-s − 4·47-s + 10·53-s + 8·55-s − 4·57-s + 8·59-s + 6·61-s + 8·65-s − 4·67-s + 6·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.78·5-s + 1/3·9-s + 0.603·11-s + 0.554·13-s + 1.03·15-s − 0.917·19-s + 1.25·23-s + 11/5·25-s + 0.192·27-s − 1.85·29-s − 1.43·31-s + 0.348·33-s + 1.64·37-s + 0.320·39-s + 0.624·41-s + 1.21·43-s + 0.596·45-s − 0.583·47-s + 1.37·53-s + 1.07·55-s − 0.529·57-s + 1.04·59-s + 0.768·61-s + 0.992·65-s − 0.488·67-s + 0.722·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4704 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.853705088\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.853705088\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 - 10 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−8.656785549668175823085882034626, −7.42488597471001832645295709361, −6.89395245158427143602186029628, −5.86726229391214216354459196326, −5.72957652207942169279900128817, −4.54665125487038859835155553843, −3.71408053957977406463284160259, −2.66946300361725093781838621415, −1.98039390334130692303329166146, −1.16269208419641589561424675896,
1.16269208419641589561424675896, 1.98039390334130692303329166146, 2.66946300361725093781838621415, 3.71408053957977406463284160259, 4.54665125487038859835155553843, 5.72957652207942169279900128817, 5.86726229391214216354459196326, 6.89395245158427143602186029628, 7.42488597471001832645295709361, 8.656785549668175823085882034626
