| L(s) = 1 | − i·2-s − 3.30i·3-s − 4-s − 3.30·6-s − 0.302i·7-s + i·8-s − 7.90·9-s − 5.30·11-s + 3.30i·12-s + 0.302i·13-s − 0.302·14-s + 16-s − 3.90i·17-s + 7.90i·18-s + 4.90·19-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 1.90i·3-s − 0.5·4-s − 1.34·6-s − 0.114i·7-s + 0.353i·8-s − 2.63·9-s − 1.59·11-s + 0.953i·12-s + 0.0839i·13-s − 0.0809·14-s + 0.250·16-s − 0.947i·17-s + 1.86i·18-s + 1.12·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4293292211\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4293292211\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 - iT \) |
| good | 3 | \( 1 + 3.30iT - 3T^{2} \) |
| 7 | \( 1 + 0.302iT - 7T^{2} \) |
| 11 | \( 1 + 5.30T + 11T^{2} \) |
| 13 | \( 1 - 0.302iT - 13T^{2} \) |
| 17 | \( 1 + 3.90iT - 17T^{2} \) |
| 19 | \( 1 - 4.90T + 19T^{2} \) |
| 29 | \( 1 + 4.60T + 29T^{2} \) |
| 31 | \( 1 - 2.90T + 31T^{2} \) |
| 37 | \( 1 - 8iT - 37T^{2} \) |
| 41 | \( 1 + 9.90T + 41T^{2} \) |
| 43 | \( 1 + 5.21iT - 43T^{2} \) |
| 47 | \( 1 - 4.60iT - 47T^{2} \) |
| 53 | \( 1 + 3.21iT - 53T^{2} \) |
| 59 | \( 1 - 10.6T + 59T^{2} \) |
| 61 | \( 1 + 6.51T + 61T^{2} \) |
| 67 | \( 1 + 4iT - 67T^{2} \) |
| 71 | \( 1 + 12.6T + 71T^{2} \) |
| 73 | \( 1 + 15.8iT - 73T^{2} \) |
| 79 | \( 1 + 14.4T + 79T^{2} \) |
| 83 | \( 1 - 3.21iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 2.69iT - 97T^{2} \) |
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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
−9.001135348752187686609585015144, −8.099103504943009642691743489913, −7.55477907886224872375086549479, −6.84461407946393813270972277500, −5.66066747521837823662955915601, −5.04265455870099342961153824576, −3.18433928996830021773184473887, −2.53431098249378248139891989219, −1.43083537192613349746032026409, −0.18563452770645763297984841007,
2.72714639332752488043557049530, 3.68607717001711619305555499634, 4.56887612037621709218770016165, 5.47567020337299612441829422783, 5.74773463783686144049757919619, 7.29513901709794789687278194061, 8.264265786806196725662724122607, 8.793492625018732014000268715178, 9.793205556551945726639921380302, 10.22333427995322545828295325062
