L(s) = 1 | + 1.41·5-s − 4·11-s + 4.24·13-s − 7.07·17-s + 5.65·19-s − 8·23-s − 2.99·25-s + 2·29-s + 4·37-s + 9.89·41-s + 4·43-s − 5.65·47-s + 4·53-s − 5.65·55-s + 11.3·59-s + 1.41·61-s + 6·65-s + 12·67-s − 15.5·73-s + 16·79-s + 5.65·83-s − 10.0·85-s + 7.07·89-s + 8.00·95-s + 7.07·97-s + 12.7·101-s − 5.65·103-s + ⋯ |
L(s) = 1 | + 0.632·5-s − 1.20·11-s + 1.17·13-s − 1.71·17-s + 1.29·19-s − 1.66·23-s − 0.599·25-s + 0.371·29-s + 0.657·37-s + 1.54·41-s + 0.609·43-s − 0.825·47-s + 0.549·53-s − 0.762·55-s + 1.47·59-s + 0.181·61-s + 0.744·65-s + 1.46·67-s − 1.82·73-s + 1.80·79-s + 0.620·83-s − 1.08·85-s + 0.749·89-s + 0.820·95-s + 0.717·97-s + 1.26·101-s − 0.557·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 7056 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.981750321\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.981750321\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 1.41T + 5T^{2} \) |
| 11 | \( 1 + 4T + 11T^{2} \) |
| 13 | \( 1 - 4.24T + 13T^{2} \) |
| 17 | \( 1 + 7.07T + 17T^{2} \) |
| 19 | \( 1 - 5.65T + 19T^{2} \) |
| 23 | \( 1 + 8T + 23T^{2} \) |
| 29 | \( 1 - 2T + 29T^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 - 4T + 37T^{2} \) |
| 41 | \( 1 - 9.89T + 41T^{2} \) |
| 43 | \( 1 - 4T + 43T^{2} \) |
| 47 | \( 1 + 5.65T + 47T^{2} \) |
| 53 | \( 1 - 4T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 - 1.41T + 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 15.5T + 73T^{2} \) |
| 79 | \( 1 - 16T + 79T^{2} \) |
| 83 | \( 1 - 5.65T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 - 7.07T + 97T^{2} \) |
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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
−7.969211615481351891843541775523, −7.32255808568705907177971119939, −6.32301369786361087113965678331, −5.93918724800471733548830556486, −5.21644729068976999674154883021, −4.34696937816104427224743853008, −3.60654459229780335061141967597, −2.51951359637012108384803085445, −1.99231613645070061606911589619, −0.70026480379897881243970751324,
0.70026480379897881243970751324, 1.99231613645070061606911589619, 2.51951359637012108384803085445, 3.60654459229780335061141967597, 4.34696937816104427224743853008, 5.21644729068976999674154883021, 5.93918724800471733548830556486, 6.32301369786361087113965678331, 7.32255808568705907177971119939, 7.969211615481351891843541775523