| L(s) = 1 | − 2-s + 4-s − 1.58·5-s + 3.85·7-s − 8-s + 1.58·10-s − 6.27·11-s + 1.11·13-s − 3.85·14-s + 16-s + 1.85·17-s + 0.102·19-s − 1.58·20-s + 6.27·22-s + 2.92·23-s − 2.48·25-s − 1.11·26-s + 3.85·28-s + 6.84·29-s + 1.61·31-s − 32-s − 1.85·34-s − 6.10·35-s − 9.34·37-s − 0.102·38-s + 1.58·40-s + 4.37·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.708·5-s + 1.45·7-s − 0.353·8-s + 0.501·10-s − 1.89·11-s + 0.308·13-s − 1.02·14-s + 0.250·16-s + 0.448·17-s + 0.0234·19-s − 0.354·20-s + 1.33·22-s + 0.608·23-s − 0.497·25-s − 0.218·26-s + 0.727·28-s + 1.27·29-s + 0.290·31-s − 0.176·32-s − 0.317·34-s − 1.03·35-s − 1.53·37-s − 0.0165·38-s + 0.250·40-s + 0.683·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2034 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2034 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.132273928\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.132273928\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 113 | \( 1 - T \) |
| good | 5 | \( 1 + 1.58T + 5T^{2} \) |
| 7 | \( 1 - 3.85T + 7T^{2} \) |
| 11 | \( 1 + 6.27T + 11T^{2} \) |
| 13 | \( 1 - 1.11T + 13T^{2} \) |
| 17 | \( 1 - 1.85T + 17T^{2} \) |
| 19 | \( 1 - 0.102T + 19T^{2} \) |
| 23 | \( 1 - 2.92T + 23T^{2} \) |
| 29 | \( 1 - 6.84T + 29T^{2} \) |
| 31 | \( 1 - 1.61T + 31T^{2} \) |
| 37 | \( 1 + 9.34T + 37T^{2} \) |
| 41 | \( 1 - 4.37T + 41T^{2} \) |
| 43 | \( 1 - 8.01T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 - 7.95T + 53T^{2} \) |
| 59 | \( 1 + 4.58T + 59T^{2} \) |
| 61 | \( 1 + 8.43T + 61T^{2} \) |
| 67 | \( 1 - 1.02T + 67T^{2} \) |
| 71 | \( 1 + 4.38T + 71T^{2} \) |
| 73 | \( 1 - 7.21T + 73T^{2} \) |
| 79 | \( 1 + 10.9T + 79T^{2} \) |
| 83 | \( 1 - 9.74T + 83T^{2} \) |
| 89 | \( 1 + 7.48T + 89T^{2} \) |
| 97 | \( 1 - 19.3T + 97T^{2} \) |
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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
−8.815709482593205414795682309893, −8.346735189921634452185872383566, −7.60906683269431550626185147997, −7.34542134674310262756559492411, −5.89287329297084624584372047266, −5.13679044515623003316498035867, −4.34417755855592274551879034214, −3.05929986827262326701318348278, −2.10504923270919225009492562128, −0.78765118015390625344637702604,
0.78765118015390625344637702604, 2.10504923270919225009492562128, 3.05929986827262326701318348278, 4.34417755855592274551879034214, 5.13679044515623003316498035867, 5.89287329297084624584372047266, 7.34542134674310262756559492411, 7.60906683269431550626185147997, 8.346735189921634452185872383566, 8.815709482593205414795682309893
