| L(s) = 1 | + 3.46i·7-s − 3.16·11-s + 4.47·13-s − 6.32·17-s + (2 − 3.87i)19-s + 5.47i·23-s + 5·25-s − 1.41·29-s − 8.94·31-s + 4.47·37-s + 2.44i·41-s + 5.47i·47-s − 4.99·49-s − 4.24·53-s + 4.89i·59-s + ⋯ |
| L(s) = 1 | + 1.30i·7-s − 0.953·11-s + 1.24·13-s − 1.53·17-s + (0.458 − 0.888i)19-s + 1.14i·23-s + 25-s − 0.262·29-s − 1.60·31-s + 0.735·37-s + 0.382i·41-s + 0.798i·47-s − 0.714·49-s − 0.582·53-s + 0.637i·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0469i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5472 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.998 + 0.0469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5000082931\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5000082931\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 19 | \( 1 + (-2 + 3.87i)T \) |
| good | 5 | \( 1 - 5T^{2} \) |
| 7 | \( 1 - 3.46iT - 7T^{2} \) |
| 11 | \( 1 + 3.16T + 11T^{2} \) |
| 13 | \( 1 - 4.47T + 13T^{2} \) |
| 17 | \( 1 + 6.32T + 17T^{2} \) |
| 23 | \( 1 - 5.47iT - 23T^{2} \) |
| 29 | \( 1 + 1.41T + 29T^{2} \) |
| 31 | \( 1 + 8.94T + 31T^{2} \) |
| 37 | \( 1 - 4.47T + 37T^{2} \) |
| 41 | \( 1 - 2.44iT - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 - 5.47iT - 47T^{2} \) |
| 53 | \( 1 + 4.24T + 53T^{2} \) |
| 59 | \( 1 - 4.89iT - 59T^{2} \) |
| 61 | \( 1 + 10.3iT - 61T^{2} \) |
| 67 | \( 1 - 7.74iT - 67T^{2} \) |
| 71 | \( 1 - 11.3T + 71T^{2} \) |
| 73 | \( 1 + 12T + 73T^{2} \) |
| 79 | \( 1 + 4.47T + 79T^{2} \) |
| 83 | \( 1 + 15.8T + 83T^{2} \) |
| 89 | \( 1 - 17.1iT - 89T^{2} \) |
| 97 | \( 1 + 15.4iT - 97T^{2} \) |
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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
−8.701765440388995766640833731860, −7.87841177972957637862627690102, −7.08380416556804737782880953982, −6.30408240056260022479108116222, −5.58770254032385417217990476296, −5.07846909588317919868640330350, −4.12818447130442495122728322098, −3.07740557281922103791471265452, −2.47257358111489877792102900408, −1.46772068458835493163531447343,
0.13094195380851589213540691652, 1.25495843948575448013754011928, 2.33195001517867849278065147089, 3.40849430118202896017963163542, 4.07018947792038667893524335293, 4.76181385366633025718131094597, 5.67677719752074971081587309726, 6.46733590585290044785539725482, 7.11107891233631403936786155260, 7.74236500030974708438860541397
