| L(s) = 1 | + 1.56i·2-s + i·3-s − 0.438·4-s − 1.56·6-s + 2.43i·8-s − 9-s − 2.56·11-s − 0.438i·12-s − 4.56i·13-s − 4.68·16-s + i·17-s − 1.56i·18-s − 7.68·19-s − 4i·22-s + 6.56i·23-s − 2.43·24-s + ⋯ |
| L(s) = 1 | + 1.10i·2-s + 0.577i·3-s − 0.219·4-s − 0.637·6-s + 0.862i·8-s − 0.333·9-s − 0.772·11-s − 0.126i·12-s − 1.26i·13-s − 1.17·16-s + 0.242i·17-s − 0.368i·18-s − 1.76·19-s − 0.852i·22-s + 1.36i·23-s − 0.497·24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1275 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.6909336387\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6909336387\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - iT \) |
| 5 | \( 1 \) |
| 17 | \( 1 - iT \) |
| good | 2 | \( 1 - 1.56iT - 2T^{2} \) |
| 7 | \( 1 - 7T^{2} \) |
| 11 | \( 1 + 2.56T + 11T^{2} \) |
| 13 | \( 1 + 4.56iT - 13T^{2} \) |
| 19 | \( 1 + 7.68T + 19T^{2} \) |
| 23 | \( 1 - 6.56iT - 23T^{2} \) |
| 29 | \( 1 + 8.24T + 29T^{2} \) |
| 31 | \( 1 + 5.12T + 31T^{2} \) |
| 37 | \( 1 - 3.12iT - 37T^{2} \) |
| 41 | \( 1 - 0.561T + 41T^{2} \) |
| 43 | \( 1 - 7.68iT - 43T^{2} \) |
| 47 | \( 1 + 2.87iT - 47T^{2} \) |
| 53 | \( 1 - 4.24iT - 53T^{2} \) |
| 59 | \( 1 - 1.12T + 59T^{2} \) |
| 61 | \( 1 - 0.876T + 61T^{2} \) |
| 67 | \( 1 - 4iT - 67T^{2} \) |
| 71 | \( 1 - 10.2T + 71T^{2} \) |
| 73 | \( 1 + 4.24iT - 73T^{2} \) |
| 79 | \( 1 + 15.3T + 79T^{2} \) |
| 83 | \( 1 - 9.12iT - 83T^{2} \) |
| 89 | \( 1 + 7.12T + 89T^{2} \) |
| 97 | \( 1 + 11.1iT - 97T^{2} \) |
| 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
−10.23367824129197947115662706876, −9.223041110817588681145474709688, −8.367442379353291575692137521517, −7.79096228369197126552547975811, −7.02116406124495562132132537116, −5.80333356931909860716390851659, −5.57243779144639184091141380983, −4.48814244387791944789496621429, −3.32630342623306722659640235768, −2.13944314295624875286127210567,
0.25618350132553105471179863321, 1.94752214393187389101849916934, 2.35723994159235203935579024573, 3.70551213182834202488051584912, 4.52806790893389124472089163577, 5.84597676705853725963340117354, 6.76531278974637797672536477174, 7.35324684485269457984434520195, 8.556505124838949590807427561348, 9.165450060305715111342866468481
