L(s) = 1 | − 2.82·5-s − 2·11-s + 1.41·17-s + 7.07·19-s − 4·23-s + 3.00·25-s − 2·29-s − 8.48·31-s + 10·37-s + 9.89·41-s − 2·43-s + 2.82·47-s + 2·53-s + 5.65·55-s − 1.41·59-s + 2.82·61-s − 12·67-s − 12·71-s − 1.41·73-s + 4·79-s + 9.89·83-s − 4.00·85-s + 7.07·89-s − 20.0·95-s + 9.89·97-s − 8.48·101-s − 2.82·103-s + ⋯ |
L(s) = 1 | − 1.26·5-s − 0.603·11-s + 0.342·17-s + 1.62·19-s − 0.834·23-s + 0.600·25-s − 0.371·29-s − 1.52·31-s + 1.64·37-s + 1.54·41-s − 0.304·43-s + 0.412·47-s + 0.274·53-s + 0.762·55-s − 0.184·59-s + 0.362·61-s − 1.46·67-s − 1.42·71-s − 0.165·73-s + 0.450·79-s + 1.08·83-s − 0.433·85-s + 0.749·89-s − 2.05·95-s + 1.00·97-s − 0.844·101-s − 0.278·103-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
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + 2.82T + 5T^{2} \) |
| 11 | \( 1 + 2T + 11T^{2} \) |
| 13 | \( 1 + 13T^{2} \) |
| 17 | \( 1 - 1.41T + 17T^{2} \) |
| 19 | \( 1 - 7.07T + 19T^{2} \) |
| 23 | \( 1 + 4T + 23T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 + 8.48T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 9.89T + 41T^{2} \) |
| 43 | \( 1 + 2T + 43T^{2} \) |
| 47 | \( 1 - 2.82T + 47T^{2} \) |
| 53 | \( 1 - 2T + 53T^{2} \) |
| 59 | \( 1 + 1.41T + 59T^{2} \) |
| 61 | \( 1 - 2.82T + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + 1.41T + 73T^{2} \) |
| 79 | \( 1 - 4T + 79T^{2} \) |
| 83 | \( 1 - 9.89T + 83T^{2} \) |
| 89 | \( 1 - 7.07T + 89T^{2} \) |
| 97 | \( 1 - 9.89T + 97T^{2} \) |
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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
−7.63419421206520839938148304363, −7.26829397609381102579143533808, −6.07228874543160584398886233676, −5.50747150414252939006727864963, −4.64617879179855531935761468529, −3.90227145086373314331756027215, −3.29573261595275243247692281225, −2.40265991502594411286578241085, −1.10626012370763942716280699843, 0,
1.10626012370763942716280699843, 2.40265991502594411286578241085, 3.29573261595275243247692281225, 3.90227145086373314331756027215, 4.64617879179855531935761468529, 5.50747150414252939006727864963, 6.07228874543160584398886233676, 7.26829397609381102579143533808, 7.63419421206520839938148304363