| L(s) = 1 | − 8·4-s − 5·5-s + 7·7-s − 42·11-s + 20·13-s + 64·16-s − 66·17-s + 38·19-s + 40·20-s − 12·23-s + 25·25-s − 56·28-s + 258·29-s + 146·31-s − 35·35-s + 434·37-s + 282·41-s + 20·43-s + 336·44-s + 72·47-s + 49·49-s − 160·52-s − 336·53-s + 210·55-s + 360·59-s − 682·61-s − 512·64-s + ⋯ |
| L(s) = 1 | − 4-s − 0.447·5-s + 0.377·7-s − 1.15·11-s + 0.426·13-s + 16-s − 0.941·17-s + 0.458·19-s + 0.447·20-s − 0.108·23-s + 1/5·25-s − 0.377·28-s + 1.65·29-s + 0.845·31-s − 0.169·35-s + 1.92·37-s + 1.07·41-s + 0.0709·43-s + 1.15·44-s + 0.223·47-s + 1/7·49-s − 0.426·52-s − 0.870·53-s + 0.514·55-s + 0.794·59-s − 1.43·61-s − 64-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.160452099\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.160452099\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + p T \) |
| 7 | \( 1 - p T \) |
| good | 2 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + 42 T + p^{3} T^{2} \) |
| 13 | \( 1 - 20 T + p^{3} T^{2} \) |
| 17 | \( 1 + 66 T + p^{3} T^{2} \) |
| 19 | \( 1 - 2 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 12 T + p^{3} T^{2} \) |
| 29 | \( 1 - 258 T + p^{3} T^{2} \) |
| 31 | \( 1 - 146 T + p^{3} T^{2} \) |
| 37 | \( 1 - 434 T + p^{3} T^{2} \) |
| 41 | \( 1 - 282 T + p^{3} T^{2} \) |
| 43 | \( 1 - 20 T + p^{3} T^{2} \) |
| 47 | \( 1 - 72 T + p^{3} T^{2} \) |
| 53 | \( 1 + 336 T + p^{3} T^{2} \) |
| 59 | \( 1 - 360 T + p^{3} T^{2} \) |
| 61 | \( 1 + 682 T + p^{3} T^{2} \) |
| 67 | \( 1 - 812 T + p^{3} T^{2} \) |
| 71 | \( 1 + 810 T + p^{3} T^{2} \) |
| 73 | \( 1 + 124 T + p^{3} T^{2} \) |
| 79 | \( 1 - 1136 T + p^{3} T^{2} \) |
| 83 | \( 1 + 156 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1038 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1208 T + p^{3} T^{2} \) |
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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
−11.15286715666112145620963228344, −10.31153961233342374182773609078, −9.280789191558558104943270885003, −8.295523579711364497311780079404, −7.72871184383678397496126096351, −6.21760747313520108846720695270, −4.97444269200799989663493631341, −4.24540220067398550921794667440, −2.78318453049259369358140902160, −0.74970079008515279609022174604,
0.74970079008515279609022174604, 2.78318453049259369358140902160, 4.24540220067398550921794667440, 4.97444269200799989663493631341, 6.21760747313520108846720695270, 7.72871184383678397496126096351, 8.295523579711364497311780079404, 9.280789191558558104943270885003, 10.31153961233342374182773609078, 11.15286715666112145620963228344
