| L(s) = 1 | + i·2-s − i·3-s − 4-s − 0.500i·5-s + 6-s − i·8-s − 9-s + 0.500·10-s + (0.278 + 3.30i)11-s + i·12-s − 0.920·13-s − 0.500·15-s + 16-s + 3.39·17-s − i·18-s − 6.51·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.223i·5-s + 0.408·6-s − 0.353i·8-s − 0.333·9-s + 0.158·10-s + (0.0839 + 0.996i)11-s + 0.288i·12-s − 0.255·13-s − 0.129·15-s + 0.250·16-s + 0.822·17-s − 0.235i·18-s − 1.49·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0731i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0731i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3718881793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3718881793\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - iT \) |
| 3 | \( 1 + iT \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-0.278 - 3.30i)T \) |
| good | 5 | \( 1 + 0.500iT - 5T^{2} \) |
| 13 | \( 1 + 0.920T + 13T^{2} \) |
| 17 | \( 1 - 3.39T + 17T^{2} \) |
| 19 | \( 1 + 6.51T + 19T^{2} \) |
| 23 | \( 1 + 0.975T + 23T^{2} \) |
| 29 | \( 1 - 4.69iT - 29T^{2} \) |
| 31 | \( 1 + 8.31iT - 31T^{2} \) |
| 37 | \( 1 + 5.56T + 37T^{2} \) |
| 41 | \( 1 + 3.99T + 41T^{2} \) |
| 43 | \( 1 - 0.666iT - 43T^{2} \) |
| 47 | \( 1 - 5.52iT - 47T^{2} \) |
| 53 | \( 1 + 7.44T + 53T^{2} \) |
| 59 | \( 1 + 11.7iT - 59T^{2} \) |
| 61 | \( 1 + 4.02T + 61T^{2} \) |
| 67 | \( 1 + 2.31T + 67T^{2} \) |
| 71 | \( 1 + 9.35T + 71T^{2} \) |
| 73 | \( 1 + 1.70T + 73T^{2} \) |
| 79 | \( 1 - 8.66iT - 79T^{2} \) |
| 83 | \( 1 - 5.60T + 83T^{2} \) |
| 89 | \( 1 - 14.3iT - 89T^{2} \) |
| 97 | \( 1 - 10.5iT - 97T^{2} \) |
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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
−8.871445961577443906738270445801, −8.089282624889202646060887571423, −7.54662831944903187151773826656, −6.74410960433842096067166808565, −6.23999553259826482730995289499, −5.22898729939500849989110214467, −4.61732566775929002305352642042, −3.64517688815545189377082082651, −2.42164253739698892750517654093, −1.38515850696048445782302001452,
0.11446116248408329279764863452, 1.55841998891937890990877012919, 2.79496456237269106096530959490, 3.39581003443505520521749314782, 4.27642834431308152844498471209, 5.07194102809231513983018390606, 5.89974021831785442247242504031, 6.68371268100386551937476666704, 7.73664441152915399182376361899, 8.737161003711124831099658330284
