| L(s) = 1 | + 3·3-s + 5·5-s + 22.0·7-s + 9·9-s − 51.4·11-s − 26.6·13-s + 15·15-s − 31.2·17-s + 134.·19-s + 66.2·21-s + 23·23-s + 25·25-s + 27·27-s − 300.·29-s + 300.·31-s − 154.·33-s + 110.·35-s + 150.·37-s − 80.0·39-s + 120.·41-s + 365.·43-s + 45·45-s + 90.3·47-s + 144.·49-s − 93.6·51-s + 487.·53-s − 257.·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s + 1.19·7-s + 0.333·9-s − 1.41·11-s − 0.569·13-s + 0.258·15-s − 0.445·17-s + 1.62·19-s + 0.688·21-s + 0.208·23-s + 0.200·25-s + 0.192·27-s − 1.92·29-s + 1.74·31-s − 0.814·33-s + 0.533·35-s + 0.669·37-s − 0.328·39-s + 0.459·41-s + 1.29·43-s + 0.149·45-s + 0.280·47-s + 0.421·49-s − 0.257·51-s + 1.26·53-s − 0.631·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1380 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.297363546\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.297363546\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 3T \) |
| 5 | \( 1 - 5T \) |
| 23 | \( 1 - 23T \) |
| good | 7 | \( 1 - 22.0T + 343T^{2} \) |
| 11 | \( 1 + 51.4T + 1.33e3T^{2} \) |
| 13 | \( 1 + 26.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 31.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 134.T + 6.85e3T^{2} \) |
| 29 | \( 1 + 300.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 300.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 150.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 120.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 365.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 90.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 487.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 609.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 9.06T + 2.26e5T^{2} \) |
| 67 | \( 1 - 392.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 58.1T + 3.57e5T^{2} \) |
| 73 | \( 1 - 1.02e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 611.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 170.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.35e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.48e3T + 9.12e5T^{2} \) |
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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
−9.290058747529421544566144419043, −8.223601401709342072141433510130, −7.74563282752575470381081730995, −7.04552932630349166764606594043, −5.59152078337101181923281938400, −5.13166141644942637300321759821, −4.14265790697619159021986433520, −2.78742443648080130269788392498, −2.15614797659674967454649826811, −0.897093793225803427819721270646,
0.897093793225803427819721270646, 2.15614797659674967454649826811, 2.78742443648080130269788392498, 4.14265790697619159021986433520, 5.13166141644942637300321759821, 5.59152078337101181923281938400, 7.04552932630349166764606594043, 7.74563282752575470381081730995, 8.223601401709342072141433510130, 9.290058747529421544566144419043
