L(s) = 1 | − 3.87i·5-s − 3.87i·11-s + 3.46i·13-s + 4·19-s − 7.74i·23-s − 10.0·25-s − 6.70·29-s − 31-s + 4·37-s − 7.74i·41-s + 6.92i·43-s − 13.4·47-s + 6.70·53-s − 15.0·55-s − 6.70·59-s + ⋯ |
L(s) = 1 | − 1.73i·5-s − 1.16i·11-s + 0.960i·13-s + 0.917·19-s − 1.61i·23-s − 2.00·25-s − 1.24·29-s − 0.179·31-s + 0.657·37-s − 1.20i·41-s + 1.05i·43-s − 1.95·47-s + 0.921·53-s − 2.02·55-s − 0.873·59-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.944 - 0.327i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.9890785989\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9890785989\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
good | 5 | \( 1 + 3.87iT - 5T^{2} \) |
| 11 | \( 1 + 3.87iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + 7.74iT - 23T^{2} \) |
| 29 | \( 1 + 6.70T + 29T^{2} \) |
| 31 | \( 1 + T + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 + 7.74iT - 41T^{2} \) |
| 43 | \( 1 - 6.92iT - 43T^{2} \) |
| 47 | \( 1 + 13.4T + 47T^{2} \) |
| 53 | \( 1 - 6.70T + 53T^{2} \) |
| 59 | \( 1 + 6.70T + 59T^{2} \) |
| 61 | \( 1 + 10.3iT - 61T^{2} \) |
| 67 | \( 1 - 6.92iT - 67T^{2} \) |
| 71 | \( 1 - 7.74iT - 71T^{2} \) |
| 73 | \( 1 + 6.92iT - 73T^{2} \) |
| 79 | \( 1 - 12.1iT - 79T^{2} \) |
| 83 | \( 1 + 6.70T + 83T^{2} \) |
| 89 | \( 1 + 7.74iT - 89T^{2} \) |
| 97 | \( 1 + 5.19iT - 97T^{2} \) |
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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
−7.75732881414341297770290501540, −6.82262999808522883934829963270, −5.98687150395457914478296041841, −5.42027437219794824652123258076, −4.68115378228741512986835231101, −4.10755386262384893302972959422, −3.20988357544793288953886458600, −2.02212308629211435461448464649, −1.13749648477910573252393607553, −0.24753622179303892227757987686,
1.53504394174330797523206353312, 2.43026732656221220935317623142, 3.26149578250211477175875619857, 3.68812297127066742060218331275, 4.85201633656257562332589149150, 5.63249855836941697766153434085, 6.26484611938309083917128957661, 7.14940514245963890926321649312, 7.47750384734131465577254757325, 7.948811663819704835177147092899