| L(s) = 1 | − i·3-s + (−0.131 + 2.23i)5-s − 5.13i·7-s − 9-s − 1.58·11-s + 2.13i·13-s + (2.23 + 0.131i)15-s + 7.84i·17-s + 2.19·19-s − 5.13·21-s − 5.33i·23-s + (−4.96 − 0.585i)25-s + i·27-s + 0.166·29-s − 1.09·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + (−0.0586 + 0.998i)5-s − 1.94i·7-s − 0.333·9-s − 0.479·11-s + 0.592i·13-s + (0.576 + 0.0338i)15-s + 1.90i·17-s + 0.502·19-s − 1.12·21-s − 1.11i·23-s + (−0.993 − 0.117i)25-s + 0.192i·27-s + 0.0308·29-s − 0.196·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0586 - 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0586 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.8536742269\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8536742269\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.131 - 2.23i)T \) |
| 67 | \( 1 - iT \) |
| good | 7 | \( 1 + 5.13iT - 7T^{2} \) |
| 11 | \( 1 + 1.58T + 11T^{2} \) |
| 13 | \( 1 - 2.13iT - 13T^{2} \) |
| 17 | \( 1 - 7.84iT - 17T^{2} \) |
| 19 | \( 1 - 2.19T + 19T^{2} \) |
| 23 | \( 1 + 5.33iT - 23T^{2} \) |
| 29 | \( 1 - 0.166T + 29T^{2} \) |
| 31 | \( 1 + 1.09T + 31T^{2} \) |
| 37 | \( 1 + 2.96iT - 37T^{2} \) |
| 41 | \( 1 + 10.9T + 41T^{2} \) |
| 43 | \( 1 - 3.88iT - 43T^{2} \) |
| 47 | \( 1 - 6.39iT - 47T^{2} \) |
| 53 | \( 1 - 11.5iT - 53T^{2} \) |
| 59 | \( 1 + 7.22T + 59T^{2} \) |
| 61 | \( 1 - 9.90T + 61T^{2} \) |
| 71 | \( 1 - 11.2T + 71T^{2} \) |
| 73 | \( 1 + 6.60iT - 73T^{2} \) |
| 79 | \( 1 + 7.35T + 79T^{2} \) |
| 83 | \( 1 - 7.97iT - 83T^{2} \) |
| 89 | \( 1 - 10.6T + 89T^{2} \) |
| 97 | \( 1 + 1.21iT - 97T^{2} \) |
| 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
−8.247883917648804285368104456151, −7.82096782352990309523658115293, −7.09881342918932097238612297226, −6.59516323767794044942946794475, −5.99338499086314948423939780983, −4.69241184550775229654926946510, −3.91613239544179314503020770447, −3.29661194637920161297702414218, −2.12892935417954263671209654876, −1.13952389723449340987090834096,
0.25475387460855700394880605040, 1.83884156800205937456873951019, 2.79536770646104298900164799617, 3.50193349932951211539150781865, 4.88338546481758516643239802142, 5.32991308932702433469805270417, 5.52521481705662553303669093505, 6.75525318384268422346299164036, 7.80035143218804793091860203224, 8.446587713257306228801625032995
