| L(s) = 1 | + 0.613i·3-s + 2.23·5-s + 2.64i·7-s + 2.62·9-s + 5.55i·11-s − 1.06·13-s + 1.37i·15-s + 5.75·17-s − 1.62·21-s + 5.00·25-s + 3.45i·27-s − 9.62·29-s − 3.40·33-s + 5.91i·35-s − 0.652i·39-s + ⋯ |
| L(s) = 1 | + 0.354i·3-s + 0.999·5-s + 0.999i·7-s + 0.874·9-s + 1.67i·11-s − 0.294·13-s + 0.354i·15-s + 1.39·17-s − 0.354·21-s + 1.00·25-s + 0.664i·27-s − 1.78·29-s − 0.593·33-s + 0.999i·35-s − 0.104i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.305763903\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.305763903\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 2.23T \) |
| 7 | \( 1 - 2.64iT \) |
| good | 3 | \( 1 - 0.613iT - 3T^{2} \) |
| 11 | \( 1 - 5.55iT - 11T^{2} \) |
| 13 | \( 1 + 1.06T + 13T^{2} \) |
| 17 | \( 1 - 5.75T + 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 + 9.62T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + 43T^{2} \) |
| 47 | \( 1 + 13.6iT - 47T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 - 11.8iT - 71T^{2} \) |
| 73 | \( 1 + 13.4T + 73T^{2} \) |
| 79 | \( 1 - 9.74iT - 79T^{2} \) |
| 83 | \( 1 + 15.8iT - 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 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
−9.431806042247392006657498832204, −8.687293598294189450676275099195, −7.47804022185059655244544806823, −7.04086571031351284232265521981, −5.89885632295414810876719174509, −5.28846067715007138699662314081, −4.58530725623588841998053720540, −3.45623074171013704168866921721, −2.22511007179534896046565364104, −1.62139850097163927499754755689,
0.832008694225358354106338651092, 1.65508066099385547718730629210, 3.02993955418780047744115626317, 3.82292084863837207629945692964, 4.95050659229405292903155905221, 5.85416614072983400101025159406, 6.38757878092911558330690343106, 7.45403346923906765526141778542, 7.80559331454455585795386679416, 8.961782405962441939116896668780
