L(s) = 1 | + i·7-s − 4.24i·11-s + 6i·13-s + 4.24·23-s − 5·25-s + 4.24·29-s + 4i·31-s − 6i·37-s + 8.48i·41-s − 6·43-s + 8.48·47-s − 49-s − 12.7·53-s + 8.48i·59-s − 6i·61-s + ⋯ |
L(s) = 1 | + 0.377i·7-s − 1.27i·11-s + 1.66i·13-s + 0.884·23-s − 25-s + 0.787·29-s + 0.718i·31-s − 0.986i·37-s + 1.32i·41-s − 0.914·43-s + 1.23·47-s − 0.142·49-s − 1.74·53-s + 1.10i·59-s − 0.768i·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.495257923\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.495257923\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - iT \) |
good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 + 4.24iT - 11T^{2} \) |
| 13 | \( 1 - 6iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 + 19T^{2} \) |
| 23 | \( 1 - 4.24T + 23T^{2} \) |
| 29 | \( 1 - 4.24T + 29T^{2} \) |
| 31 | \( 1 - 4iT - 31T^{2} \) |
| 37 | \( 1 + 6iT - 37T^{2} \) |
| 41 | \( 1 - 8.48iT - 41T^{2} \) |
| 43 | \( 1 + 6T + 43T^{2} \) |
| 47 | \( 1 - 8.48T + 47T^{2} \) |
| 53 | \( 1 + 12.7T + 53T^{2} \) |
| 59 | \( 1 - 8.48iT - 59T^{2} \) |
| 61 | \( 1 + 6iT - 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 - 4.24T + 71T^{2} \) |
| 73 | \( 1 + 2T + 73T^{2} \) |
| 79 | \( 1 - 10iT - 79T^{2} \) |
| 83 | \( 1 - 16.9iT - 83T^{2} \) |
| 89 | \( 1 - 8.48iT - 89T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−8.582126208714631061290678870827, −8.055724235078652438779737146408, −6.97758220456534074887444198307, −6.45728967754517725464777800557, −5.69141735556805144807283262511, −4.86784729279691208672959611053, −4.00705067315927215019143456888, −3.16931585724086181994128067915, −2.21319651930375439917683443574, −1.11474261491534579866124691249,
0.46918899259827702462309735149, 1.72993651638594484156621580269, 2.77943286531179220805733196495, 3.61746058591613972159331306048, 4.58598802393403868999780784986, 5.20548079281903520116489852706, 6.06525497686410357034758676150, 6.91973077240824308686450639651, 7.60895450742240635917104821739, 8.102515148237133141706101506263