| L(s) = 1 | + i·2-s + i·3-s − 4-s + 4.30i·5-s − 6-s − i·8-s − 9-s − 4.30·10-s + (2.11 + 2.55i)11-s − i·12-s + 1.00·13-s − 4.30·15-s + 16-s + 3.33·17-s − i·18-s − 3.23·19-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + 0.577i·3-s − 0.5·4-s + 1.92i·5-s − 0.408·6-s − 0.353i·8-s − 0.333·9-s − 1.36·10-s + (0.637 + 0.770i)11-s − 0.288i·12-s + 0.277·13-s − 1.11·15-s + 0.250·16-s + 0.809·17-s − 0.235i·18-s − 0.741·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0225 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0225 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9518275769\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9518275769\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-2.11 - 2.55i)T \) |
| good | 5 | \( 1 - 4.30iT - 5T^{2} \) |
| 13 | \( 1 - 1.00T + 13T^{2} \) |
| 17 | \( 1 - 3.33T + 17T^{2} \) |
| 19 | \( 1 + 3.23T + 19T^{2} \) |
| 23 | \( 1 + 5.72T + 23T^{2} \) |
| 29 | \( 1 + 7.74iT - 29T^{2} \) |
| 31 | \( 1 - 3.20iT - 31T^{2} \) |
| 37 | \( 1 + 2.52T + 37T^{2} \) |
| 41 | \( 1 - 1.45T + 41T^{2} \) |
| 43 | \( 1 - 4.14iT - 43T^{2} \) |
| 47 | \( 1 - 3.20iT - 47T^{2} \) |
| 53 | \( 1 + 11.3T + 53T^{2} \) |
| 59 | \( 1 + 3.44iT - 59T^{2} \) |
| 61 | \( 1 + 14.8T + 61T^{2} \) |
| 67 | \( 1 + 0.330T + 67T^{2} \) |
| 71 | \( 1 - 2.84T + 71T^{2} \) |
| 73 | \( 1 - 14.8T + 73T^{2} \) |
| 79 | \( 1 - 16.0iT - 79T^{2} \) |
| 83 | \( 1 - 1.75T + 83T^{2} \) |
| 89 | \( 1 + 2.78iT - 89T^{2} \) |
| 97 | \( 1 - 12.3iT - 97T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.518806283417938351402042553813, −8.138772374772387902797815587438, −7.75244300849174859337212911630, −6.78941751124475854805895411996, −6.34649721418432793814054445257, −5.71241510223162277788579074021, −4.46376374750063622088299761651, −3.82094315411355104755539926572, −3.02614788064498737620400748201, −1.95596242184774366048249181858,
0.29983057353607263992314778028, 1.27348831162938794860104873539, 1.90583004164163440106523263931, 3.37552483734430184654432002239, 4.11228306516959441024794084857, 4.97996114203919013320721138020, 5.70456730306559180529227872846, 6.37910751096517917758398001467, 7.75537925905481370657406913992, 8.216199306221964258997096939489
