| L(s) = 1 | − 22·5-s + 92·13-s − 104·17-s + 359·25-s − 130·29-s − 396·37-s − 472·41-s − 343·49-s + 518·53-s + 468·61-s − 2.02e3·65-s − 1.09e3·73-s + 2.28e3·85-s − 176·89-s + 594·97-s + 598·101-s − 1.46e3·109-s − 1.32e3·113-s + ⋯ |
| L(s) = 1 | − 1.96·5-s + 1.96·13-s − 1.48·17-s + 2.87·25-s − 0.832·29-s − 1.75·37-s − 1.79·41-s − 49-s + 1.34·53-s + 0.982·61-s − 3.86·65-s − 1.76·73-s + 2.91·85-s − 0.209·89-s + 0.621·97-s + 0.589·101-s − 1.28·109-s − 1.10·113-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
| \(L(2)\) |
\(\approx\) |
\(0.8641770387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8641770387\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 22 T + p^{3} T^{2} \) |
| 7 | \( 1 + p^{3} T^{2} \) |
| 11 | \( 1 + p^{3} T^{2} \) |
| 13 | \( 1 - 92 T + p^{3} T^{2} \) |
| 17 | \( 1 + 104 T + p^{3} T^{2} \) |
| 19 | \( 1 + p^{3} T^{2} \) |
| 23 | \( 1 + p^{3} T^{2} \) |
| 29 | \( 1 + 130 T + p^{3} T^{2} \) |
| 31 | \( 1 + p^{3} T^{2} \) |
| 37 | \( 1 + 396 T + p^{3} T^{2} \) |
| 41 | \( 1 + 472 T + p^{3} T^{2} \) |
| 43 | \( 1 + p^{3} T^{2} \) |
| 47 | \( 1 + p^{3} T^{2} \) |
| 53 | \( 1 - 518 T + p^{3} T^{2} \) |
| 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 | \( 1 + 176 T + p^{3} T^{2} \) |
| 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
−8.573065004110877097500668596857, −8.067841219326097425074021878722, −7.02551861936373647791858835461, −6.63674372222561961232827868139, −5.41142789263618143862438458209, −4.39492976990999965621362891052, −3.78724013653643448209542827551, −3.19103615745325118550631402959, −1.64930981923357074881180619576, −0.42089949659643194919601469717,
0.42089949659643194919601469717, 1.64930981923357074881180619576, 3.19103615745325118550631402959, 3.78724013653643448209542827551, 4.39492976990999965621362891052, 5.41142789263618143862438458209, 6.63674372222561961232827868139, 7.02551861936373647791858835461, 8.067841219326097425074021878722, 8.573065004110877097500668596857
