L(s) = 1 | + 0.933·5-s + (−2.45 − 0.981i)7-s − 1.75i·11-s + 4.20i·13-s − 0.231·17-s − 5.00i·19-s + 2.61i·23-s − 4.12·25-s + 3.77i·29-s − 0.840i·31-s + (−2.29 − 0.915i)35-s + 3.01·37-s − 8.98·41-s − 2.40·43-s − 1.03·47-s + ⋯ |
L(s) = 1 | + 0.417·5-s + (−0.928 − 0.371i)7-s − 0.529i·11-s + 1.16i·13-s − 0.0560·17-s − 1.14i·19-s + 0.545i·23-s − 0.825·25-s + 0.700i·29-s − 0.150i·31-s + (−0.387 − 0.154i)35-s + 0.495·37-s − 1.40·41-s − 0.366·43-s − 0.150·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.2070896897\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2070896897\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (2.45 + 0.981i)T \) |
good | 5 | \( 1 - 0.933T + 5T^{2} \) |
| 11 | \( 1 + 1.75iT - 11T^{2} \) |
| 13 | \( 1 - 4.20iT - 13T^{2} \) |
| 17 | \( 1 + 0.231T + 17T^{2} \) |
| 19 | \( 1 + 5.00iT - 19T^{2} \) |
| 23 | \( 1 - 2.61iT - 23T^{2} \) |
| 29 | \( 1 - 3.77iT - 29T^{2} \) |
| 31 | \( 1 + 0.840iT - 31T^{2} \) |
| 37 | \( 1 - 3.01T + 37T^{2} \) |
| 41 | \( 1 + 8.98T + 41T^{2} \) |
| 43 | \( 1 + 2.40T + 43T^{2} \) |
| 47 | \( 1 + 1.03T + 47T^{2} \) |
| 53 | \( 1 + 9.04iT - 53T^{2} \) |
| 59 | \( 1 + 6.77T + 59T^{2} \) |
| 61 | \( 1 - 8.93iT - 61T^{2} \) |
| 67 | \( 1 + 1.10T + 67T^{2} \) |
| 71 | \( 1 - 7.21iT - 71T^{2} \) |
| 73 | \( 1 - 3.44iT - 73T^{2} \) |
| 79 | \( 1 + 5.29T + 79T^{2} \) |
| 83 | \( 1 + 14.8T + 83T^{2} \) |
| 89 | \( 1 + 15.0T + 89T^{2} \) |
| 97 | \( 1 - 1.63iT - 97T^{2} \) |
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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
−9.086653885442738939417346768084, −8.490495067391547219216145507431, −7.33126364452068506638169592087, −6.79679979304859132296294214051, −6.13118939957739342762928641521, −5.28314874491463492682356367052, −4.30611265649556613868448592579, −3.48662582233806518920935140893, −2.58169068481962539640815988543, −1.42690464615715694611721573375,
0.06189150180042783347037869561, 1.65518162571956985751968778646, 2.70475747734011579232518092321, 3.48149514741537629599677167358, 4.47706984796200888221536750175, 5.55405261210483466436744819919, 6.01244164135984558107299588261, 6.78468348854143113566591381868, 7.73488876905249485887412164426, 8.335629199405530184375744817487