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 | \( 1 + 9.89T + 97T^{2} \) |
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
−7.66562304355682514006027259131, −6.63476741407752544018119431696, −6.21890700748068679907332000733, −5.61705774145687170352665504992, −4.73887500195882306878324865613, −4.12919843840559892704619408804, −2.89290883292712443480801952091, −2.26480317452441476068210717346, −1.49859480313548222904571111775, 0,
1.49859480313548222904571111775, 2.26480317452441476068210717346, 2.89290883292712443480801952091, 4.12919843840559892704619408804, 4.73887500195882306878324865613, 5.61705774145687170352665504992, 6.21890700748068679907332000733, 6.63476741407752544018119431696, 7.66562304355682514006027259131