L(s) = 1 | + (−0.382 − 0.923i)5-s + i·9-s + (−1.30 − 1.30i)13-s + (−0.541 + 0.541i)17-s + (−0.707 + 0.707i)25-s − 1.41i·29-s + (−1.41 − 1.41i)37-s − 1.84i·41-s + (0.923 − 0.382i)45-s + (−1 + i)53-s + 0.765i·61-s + (−0.707 + 1.70i)65-s + (−0.541 − 0.541i)73-s − 81-s + (0.707 + 0.292i)85-s + ⋯ |
L(s) = 1 | + (−0.382 − 0.923i)5-s + i·9-s + (−1.30 − 1.30i)13-s + (−0.541 + 0.541i)17-s + (−0.707 + 0.707i)25-s − 1.41i·29-s + (−1.41 − 1.41i)37-s − 1.84i·41-s + (0.923 − 0.382i)45-s + (−1 + i)53-s + 0.765i·61-s + (−0.707 + 1.70i)65-s + (−0.541 − 0.541i)73-s − 81-s + (0.707 + 0.292i)85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.865 + 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4858231709\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4858231709\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.382 + 0.923i)T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - iT^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + (1.30 + 1.30i)T + iT^{2} \) |
| 17 | \( 1 + (0.541 - 0.541i)T - iT^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + 1.41iT - T^{2} \) |
| 31 | \( 1 + T^{2} \) |
| 37 | \( 1 + (1.41 + 1.41i)T + iT^{2} \) |
| 41 | \( 1 + 1.84iT - T^{2} \) |
| 43 | \( 1 + iT^{2} \) |
| 47 | \( 1 + iT^{2} \) |
| 53 | \( 1 + (1 - i)T - iT^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - 0.765iT - T^{2} \) |
| 67 | \( 1 - iT^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + (0.541 + 0.541i)T + iT^{2} \) |
| 79 | \( 1 + T^{2} \) |
| 83 | \( 1 - iT^{2} \) |
| 89 | \( 1 + 0.765T + T^{2} \) |
| 97 | \( 1 + (1.30 - 1.30i)T - iT^{2} \) |
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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
−8.292046439178275321369684333409, −7.64033885625493520185345875911, −7.19385012514864520032068609670, −5.82799173530573509205237776248, −5.34214540694665673058057062088, −4.61064563405625941240034101077, −3.87354357405061775161264392689, −2.66725119416108736071871952287, −1.83402563255309843185158657600, −0.25515624994551923654888876164,
1.66954332889816148973458914029, 2.79055536947196672104633573995, 3.43088530351153404625137716452, 4.42385626864748815714567047855, 5.05678914336009139205767829226, 6.34203854242738806821476338874, 6.81280070774284042049618729705, 7.21056640490637271893770257602, 8.215792803674233635057347095560, 9.012630393855166331167768383478