L(s) = 1 | + i·3-s + i·5-s − 1.52·7-s − 9-s − 6.43·11-s − 4.50·13-s − 15-s + 1.65i·17-s + 5.57·19-s − 1.52i·21-s + (2.99 − 3.74i)23-s − 25-s − i·27-s − 2.30·29-s − 1.77i·31-s + ⋯ |
L(s) = 1 | + 0.577i·3-s + 0.447i·5-s − 0.575·7-s − 0.333·9-s − 1.93·11-s − 1.24·13-s − 0.258·15-s + 0.400i·17-s + 1.27·19-s − 0.332i·21-s + (0.623 − 0.781i)23-s − 0.200·25-s − 0.192i·27-s − 0.427·29-s − 0.318i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.988 + 0.149i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5520 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.988 + 0.149i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8283836053\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8283836053\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - iT \) |
| 5 | \( 1 - iT \) |
| 23 | \( 1 + (-2.99 + 3.74i)T \) |
good | 7 | \( 1 + 1.52T + 7T^{2} \) |
| 11 | \( 1 + 6.43T + 11T^{2} \) |
| 13 | \( 1 + 4.50T + 13T^{2} \) |
| 17 | \( 1 - 1.65iT - 17T^{2} \) |
| 19 | \( 1 - 5.57T + 19T^{2} \) |
| 29 | \( 1 + 2.30T + 29T^{2} \) |
| 31 | \( 1 + 1.77iT - 31T^{2} \) |
| 37 | \( 1 - 5.43iT - 37T^{2} \) |
| 41 | \( 1 + 2.07T + 41T^{2} \) |
| 43 | \( 1 + 4.75T + 43T^{2} \) |
| 47 | \( 1 + 2.99iT - 47T^{2} \) |
| 53 | \( 1 + 2.72iT - 53T^{2} \) |
| 59 | \( 1 + 3.99iT - 59T^{2} \) |
| 61 | \( 1 - 9.74iT - 61T^{2} \) |
| 67 | \( 1 - 3.71T + 67T^{2} \) |
| 71 | \( 1 + 2.39iT - 71T^{2} \) |
| 73 | \( 1 + 9.23T + 73T^{2} \) |
| 79 | \( 1 - 11.4T + 79T^{2} \) |
| 83 | \( 1 - 9.35T + 83T^{2} \) |
| 89 | \( 1 + 2.66iT - 89T^{2} \) |
| 97 | \( 1 + 8.66iT - 97T^{2} \) |
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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
−8.017385434215877575072291678988, −7.49278685519546524123851127491, −6.78661684993206069682001588088, −5.86599981751370124713323534298, −5.10790424669586214715586144624, −4.71474598814175792252623279113, −3.40982451638614877794772338182, −2.93474474291758621104186947731, −2.16112533332015219185335078869, −0.33009427053751723753196412835,
0.63734597319179879331817176826, 1.98041879872206863224228077622, 2.81604115807354758628022974193, 3.42745296662001879080595688140, 4.96302531132379982862354418657, 5.11286157644548658588256040056, 5.92255379046240202924752806897, 6.96607297185526196679611052377, 7.58633203716671064877054198186, 7.83999069499635780566386654148