| L(s) = 1 | − i·2-s − i·3-s − 4-s − 6-s + 2i·7-s + i·8-s + 2·9-s − 3·11-s + i·12-s − i·13-s + 2·14-s + 16-s − 8i·17-s − 2i·18-s + 2·21-s + 3i·22-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.577i·3-s − 0.5·4-s − 0.408·6-s + 0.755i·7-s + 0.353i·8-s + 0.666·9-s − 0.904·11-s + 0.288i·12-s − 0.277i·13-s + 0.534·14-s + 0.250·16-s − 1.94i·17-s − 0.471i·18-s + 0.436·21-s + 0.639i·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.222811854\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.222811854\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 5 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 3 | \( 1 + iT - 3T^{2} \) |
| 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 3T + 11T^{2} \) |
| 13 | \( 1 + iT - 13T^{2} \) |
| 17 | \( 1 + 8iT - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4iT - 23T^{2} \) |
| 31 | \( 1 + 3T + 31T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 - 2T + 41T^{2} \) |
| 43 | \( 1 + 11iT - 43T^{2} \) |
| 47 | \( 1 + 13iT - 47T^{2} \) |
| 53 | \( 1 + 11iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 + 8T + 61T^{2} \) |
| 67 | \( 1 - 12iT - 67T^{2} \) |
| 71 | \( 1 - 2T + 71T^{2} \) |
| 73 | \( 1 - 4iT - 73T^{2} \) |
| 79 | \( 1 + 15T + 79T^{2} \) |
| 83 | \( 1 - 4iT - 83T^{2} \) |
| 89 | \( 1 - 10T + 89T^{2} \) |
| 97 | \( 1 - 2iT - 97T^{2} \) |
| 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
−9.282708863658135757727897311368, −8.487515671598840071247803635051, −7.48402816835388469244948131603, −7.01138723621797215961796653114, −5.56138675528115967421074827793, −5.14278588832138914268892894165, −3.86934664072090993796106148388, −2.72117007321105440843291562410, −1.99883379181252830533277575355, −0.50934430537415501617057669376,
1.43337419471930793188795317925, 3.13815152938881594795222410751, 4.31172238911361323212801375920, 4.58615571580410205780919613967, 5.87624576083343006235451222415, 6.55377279749605764139928304380, 7.57394862267748566163786555293, 8.074362155823193822037270074564, 9.041531891321149447817572604306, 9.885830763888100270512883690511
