L(s) = 1 | − 2.23·3-s − 2.23i·5-s − 2.64i·7-s + 2.00·9-s + 5.91·11-s + 6.70i·13-s + 5.00i·15-s − 7.93·17-s + 5.91i·21-s − 5.00·25-s + 2.23·27-s + 5.91i·29-s − 13.2·33-s − 5.91·35-s − 15.0i·39-s + ⋯ |
L(s) = 1 | − 1.29·3-s − 0.999i·5-s − 0.999i·7-s + 0.666·9-s + 1.78·11-s + 1.86i·13-s + 1.29i·15-s − 1.92·17-s + 1.29i·21-s − 1.00·25-s + 0.430·27-s + 1.09i·29-s − 2.30·33-s − 0.999·35-s − 2.40i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2240 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7665649832\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7665649832\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + 2.23iT \) |
| 7 | \( 1 + 2.64iT \) |
good | 3 | \( 1 + 2.23T + 3T^{2} \) |
| 11 | \( 1 - 5.91T + 11T^{2} \) |
| 13 | \( 1 - 6.70iT - 13T^{2} \) |
| 17 | \( 1 + 7.93T + 17T^{2} \) |
| 19 | \( 1 - 19T^{2} \) |
| 23 | \( 1 + 23T^{2} \) |
| 29 | \( 1 - 5.91iT - 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 - 7.93iT - 47T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 - 12iT - 71T^{2} \) |
| 73 | \( 1 - 10.5T + 73T^{2} \) |
| 79 | \( 1 + iT - 79T^{2} \) |
| 83 | \( 1 + 8.94T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 18.5T + 97T^{2} \) |
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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
−9.065840358027298987367457247822, −8.685791478319854124441912315890, −7.23724144057431727301676185043, −6.59538664449783121153353057833, −6.26357371750901138414937098593, −4.95608975790664234993917735827, −4.37839299261390115690702475443, −3.92176308393417403645909505016, −1.81032009104188220886975299848, −0.994307585835158210342344286584,
0.38815963055334966422448012020, 2.03483184794200201888307213814, 3.08309469498656969635500520792, 4.14297935040859986320471406394, 5.16392291906027062009632282024, 6.03542824907553566759176546747, 6.34787308425668230727589250279, 7.06635121117444475526736631953, 8.195791214714011737484840457784, 8.981399668716033941262269567835