| L(s) = 1 | + (−2.12 + 2.12i)2-s − 0.999i·4-s + (−9 − 9i)7-s + (−14.8 − 14.8i)8-s + 38.1i·11-s + (63 − 63i)13-s + 38.1·14-s + 71·16-s + (−29.6 + 29.6i)17-s − 70i·19-s + (−81 − 81i)22-s + (−72.1 − 72.1i)23-s + 267. i·26-s + (−8.99 + 8.99i)28-s + 229.·29-s + ⋯ |
| L(s) = 1 | + (−0.749 + 0.749i)2-s − 0.124i·4-s + (−0.485 − 0.485i)7-s + (−0.656 − 0.656i)8-s + 1.04i·11-s + (1.34 − 1.34i)13-s + 0.728·14-s + 1.10·16-s + (−0.423 + 0.423i)17-s − 0.845i·19-s + (−0.784 − 0.784i)22-s + (−0.653 − 0.653i)23-s + 2.01i·26-s + (−0.0607 + 0.0607i)28-s + 1.46·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.920 - 0.391i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.997716 + 0.203152i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.997716 + 0.203152i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 + (2.12 - 2.12i)T - 8iT^{2} \) |
| 7 | \( 1 + (9 + 9i)T + 343iT^{2} \) |
| 11 | \( 1 - 38.1iT - 1.33e3T^{2} \) |
| 13 | \( 1 + (-63 + 63i)T - 2.19e3iT^{2} \) |
| 17 | \( 1 + (29.6 - 29.6i)T - 4.91e3iT^{2} \) |
| 19 | \( 1 + 70iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (72.1 + 72.1i)T + 1.21e4iT^{2} \) |
| 29 | \( 1 - 229.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 196T + 2.97e4T^{2} \) |
| 37 | \( 1 + (-207 - 207i)T + 5.06e4iT^{2} \) |
| 41 | \( 1 - 267. iT - 6.89e4T^{2} \) |
| 43 | \( 1 + (144 - 144i)T - 7.95e4iT^{2} \) |
| 47 | \( 1 + (-356. + 356. i)T - 1.03e5iT^{2} \) |
| 53 | \( 1 + (-224. - 224. i)T + 1.48e5iT^{2} \) |
| 59 | \( 1 - 267.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 322T + 2.26e5T^{2} \) |
| 67 | \( 1 + (-378 - 378i)T + 3.00e5iT^{2} \) |
| 71 | \( 1 + 840. iT - 3.57e5T^{2} \) |
| 73 | \( 1 + (-378 + 378i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + 488iT - 4.93e5T^{2} \) |
| 83 | \( 1 + (772. + 772. i)T + 5.71e5iT^{2} \) |
| 89 | \( 1 - 267.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (252 + 252i)T + 9.12e5iT^{2} \) |
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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
−11.91931442163951603510776390922, −10.48454355849448374207696799151, −9.892942559531989903244534911378, −8.627057240684822422003648921734, −8.001510769350798589906393532469, −6.83050332634696054167295389856, −6.14733036677759785718725324436, −4.40614853134474044487183258207, −3.02171018969889954534999084204, −0.71211678185508057331020437424,
1.02962867128183891357801609789, 2.48722408296352792147509164224, 3.86890132869337120990603772110, 5.72312242033563260886396893140, 6.46890911572913046508142907506, 8.268849704843283699012750471437, 8.926872681211197544757366460695, 9.754792429895934466945963615021, 10.82060604326316792925968070814, 11.53077657125340480119568721328
