| L(s) = 1 | − 2-s + 4-s + 1.53·5-s + 4.84·7-s − 8-s − 1.53·10-s + 3.93·11-s + 3.57·13-s − 4.84·14-s + 16-s − 1.48·17-s − 7.62·19-s + 1.53·20-s − 3.93·22-s + 5.95·23-s − 2.64·25-s − 3.57·26-s + 4.84·28-s − 9.31·29-s + 4.91·31-s − 32-s + 1.48·34-s + 7.43·35-s − 1.32·37-s + 7.62·38-s − 1.53·40-s − 3.69·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.686·5-s + 1.82·7-s − 0.353·8-s − 0.485·10-s + 1.18·11-s + 0.992·13-s − 1.29·14-s + 0.250·16-s − 0.361·17-s − 1.75·19-s + 0.343·20-s − 0.839·22-s + 1.24·23-s − 0.528·25-s − 0.701·26-s + 0.914·28-s − 1.73·29-s + 0.883·31-s − 0.176·32-s + 0.255·34-s + 1.25·35-s − 0.218·37-s + 1.23·38-s − 0.242·40-s − 0.577·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.497062315\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.497062315\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 149 | \( 1 + T \) |
| good | 5 | \( 1 - 1.53T + 5T^{2} \) |
| 7 | \( 1 - 4.84T + 7T^{2} \) |
| 11 | \( 1 - 3.93T + 11T^{2} \) |
| 13 | \( 1 - 3.57T + 13T^{2} \) |
| 17 | \( 1 + 1.48T + 17T^{2} \) |
| 19 | \( 1 + 7.62T + 19T^{2} \) |
| 23 | \( 1 - 5.95T + 23T^{2} \) |
| 29 | \( 1 + 9.31T + 29T^{2} \) |
| 31 | \( 1 - 4.91T + 31T^{2} \) |
| 37 | \( 1 + 1.32T + 37T^{2} \) |
| 41 | \( 1 + 3.69T + 41T^{2} \) |
| 43 | \( 1 - 8.73T + 43T^{2} \) |
| 47 | \( 1 + 6.61T + 47T^{2} \) |
| 53 | \( 1 + 4.60T + 53T^{2} \) |
| 59 | \( 1 + 1.97T + 59T^{2} \) |
| 61 | \( 1 + 5.88T + 61T^{2} \) |
| 67 | \( 1 - 16.0T + 67T^{2} \) |
| 71 | \( 1 - 5.26T + 71T^{2} \) |
| 73 | \( 1 + 3.32T + 73T^{2} \) |
| 79 | \( 1 - 3.87T + 79T^{2} \) |
| 83 | \( 1 - 11.0T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 - 17.1T + 97T^{2} \) |
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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
−8.043658744637270970966805295539, −7.21366052762886923960703901861, −6.43375573578020390640500965165, −5.96140019938292814720568274151, −5.03813007640532036808808237312, −4.33172352852506430519253761154, −3.54851891061237069244924534458, −2.10426336221458457618007880356, −1.80698255445826888546928782438, −0.932758106612295087073151752979,
0.932758106612295087073151752979, 1.80698255445826888546928782438, 2.10426336221458457618007880356, 3.54851891061237069244924534458, 4.33172352852506430519253761154, 5.03813007640532036808808237312, 5.96140019938292814720568274151, 6.43375573578020390640500965165, 7.21366052762886923960703901861, 8.043658744637270970966805295539
