| L(s) = 1 | + 2-s − 6·3-s − 7·4-s + 5·5-s − 6·6-s − 15·8-s + 9·9-s + 5·10-s − 44·11-s + 42·12-s − 6·13-s − 30·15-s + 41·16-s + 24·17-s + 9·18-s + 114·19-s − 35·20-s − 44·22-s − 52·23-s + 90·24-s + 25·25-s − 6·26-s + 108·27-s + 146·29-s − 30·30-s + 276·31-s + 161·32-s + ⋯ |
| L(s) = 1 | + 0.353·2-s − 1.15·3-s − 7/8·4-s + 0.447·5-s − 0.408·6-s − 0.662·8-s + 1/3·9-s + 0.158·10-s − 1.20·11-s + 1.01·12-s − 0.128·13-s − 0.516·15-s + 0.640·16-s + 0.342·17-s + 0.117·18-s + 1.37·19-s − 0.391·20-s − 0.426·22-s − 0.471·23-s + 0.765·24-s + 1/5·25-s − 0.0452·26-s + 0.769·27-s + 0.934·29-s − 0.182·30-s + 1.59·31-s + 0.889·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 245 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.9380469466\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9380469466\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 - p T \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 - T + p^{3} T^{2} \) |
| 3 | \( 1 + 2 p T + p^{3} T^{2} \) |
| 11 | \( 1 + 4 p T + p^{3} T^{2} \) |
| 13 | \( 1 + 6 T + p^{3} T^{2} \) |
| 17 | \( 1 - 24 T + p^{3} T^{2} \) |
| 19 | \( 1 - 6 p T + p^{3} T^{2} \) |
| 23 | \( 1 + 52 T + p^{3} T^{2} \) |
| 29 | \( 1 - 146 T + p^{3} T^{2} \) |
| 31 | \( 1 - 276 T + p^{3} T^{2} \) |
| 37 | \( 1 + 210 T + p^{3} T^{2} \) |
| 41 | \( 1 + 444 T + p^{3} T^{2} \) |
| 43 | \( 1 - 492 T + p^{3} T^{2} \) |
| 47 | \( 1 - 612 T + p^{3} T^{2} \) |
| 53 | \( 1 - 50 T + p^{3} T^{2} \) |
| 59 | \( 1 + 294 T + p^{3} T^{2} \) |
| 61 | \( 1 + 450 T + p^{3} T^{2} \) |
| 67 | \( 1 + 668 T + p^{3} T^{2} \) |
| 71 | \( 1 + 308 T + p^{3} T^{2} \) |
| 73 | \( 1 + 12 T + p^{3} T^{2} \) |
| 79 | \( 1 - 596 T + p^{3} T^{2} \) |
| 83 | \( 1 - 966 T + p^{3} T^{2} \) |
| 89 | \( 1 - 408 T + p^{3} T^{2} \) |
| 97 | \( 1 - 1200 T + p^{3} T^{2} \) |
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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
−11.97693236302079478916350455062, −10.56668606870212241137361683026, −10.01368956894195597180246174670, −8.801811997129755070097570128571, −7.62987393674825266121058001125, −6.15153347447503958106153449847, −5.38955411650657020820502524891, −4.69012342516492712295988571953, −2.98874167807381179399889932454, −0.70760168291128778676896138659,
0.70760168291128778676896138659, 2.98874167807381179399889932454, 4.69012342516492712295988571953, 5.38955411650657020820502524891, 6.15153347447503958106153449847, 7.62987393674825266121058001125, 8.801811997129755070097570128571, 10.01368956894195597180246174670, 10.56668606870212241137361683026, 11.97693236302079478916350455062
