| L(s) = 1 | − 4·5-s − 92·13-s − 94·17-s − 109·25-s + 284·29-s − 396·37-s − 230·41-s − 343·49-s + 572·53-s + 468·61-s + 368·65-s + 1.09e3·73-s + 376·85-s + 1.67e3·89-s − 594·97-s − 1.94e3·101-s + 1.46e3·109-s − 2.00e3·113-s + ⋯ |
| L(s) = 1 | − 0.357·5-s − 1.96·13-s − 1.34·17-s − 0.871·25-s + 1.81·29-s − 1.75·37-s − 0.876·41-s − 49-s + 1.48·53-s + 0.982·61-s + 0.702·65-s + 1.76·73-s + 0.479·85-s + 1.98·89-s − 0.621·97-s − 1.91·101-s + 1.28·109-s − 1.66·113-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2304 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9286186616\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9286186616\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 4 T + p^{3} T^{2} \) |
| 7 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 + 92 T + p^{3} T^{2} \) |
| 17 | \( 1 + 94 T + p^{3} T^{2} \) |
| 19 | \( 1 + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 - 284 T + p^{3} T^{2} \) |
| 31 | \( 1 + p^{3} T^{2} \) |
| 37 | \( 1 + 396 T + p^{3} T^{2} \) |
| 41 | \( 1 + 230 T + p^{3} T^{2} \) |
| 43 | \( 1 + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 - 572 T + p^{3} T^{2} \) |
| 59 | \( 1 + p^{3} T^{2} \) |
| 61 | \( 1 - 468 T + p^{3} T^{2} \) |
| 67 | \( 1 + p^{3} T^{2} \) |
| 71 | \( 1 + p^{3} T^{2} \) |
| 73 | \( 1 - 1098 T + p^{3} T^{2} \) |
| 79 | \( 1 + p^{3} T^{2} \) |
| 83 | \( 1 + p^{3} T^{2} \) |
| 89 | \( 1 - 1670 T + p^{3} T^{2} \) |
| 97 | \( 1 + 594 T + p^{3} T^{2} \) |
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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
−8.580689417311253752583444398602, −7.926555081265494798012822103529, −6.99116504037684746951230723644, −6.60388540994122700172288225236, −5.27449088156944435613751527355, −4.74789353111988563186271055401, −3.84370632898153048891723693856, −2.71154942358699071763442400635, −1.95827488287293059866162351913, −0.40771021240714739864867759523,
0.40771021240714739864867759523, 1.95827488287293059866162351913, 2.71154942358699071763442400635, 3.84370632898153048891723693856, 4.74789353111988563186271055401, 5.27449088156944435613751527355, 6.60388540994122700172288225236, 6.99116504037684746951230723644, 7.926555081265494798012822103529, 8.580689417311253752583444398602
