| L(s) = 1 | + 2-s + 4-s − 2.82·5-s − 2.82·7-s + 8-s − 2.82·10-s + 5.65·11-s + 6·13-s − 2.82·14-s + 16-s + 2.82·17-s − 8.48·19-s − 2.82·20-s + 5.65·22-s + 3.00·25-s + 6·26-s − 2.82·28-s − 2·29-s + 8·31-s + 32-s + 2.82·34-s + 8.00·35-s − 8.48·38-s − 2.82·40-s + 6·41-s + 2.82·43-s + 5.65·44-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.5·4-s − 1.26·5-s − 1.06·7-s + 0.353·8-s − 0.894·10-s + 1.70·11-s + 1.66·13-s − 0.755·14-s + 0.250·16-s + 0.685·17-s − 1.94·19-s − 0.632·20-s + 1.20·22-s + 0.600·25-s + 1.17·26-s − 0.534·28-s − 0.371·29-s + 1.43·31-s + 0.176·32-s + 0.485·34-s + 1.35·35-s − 1.37·38-s − 0.447·40-s + 0.937·41-s + 0.431·43-s + 0.852·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9522 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.522224353\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.522224353\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 23 | \( 1 \) |
| good | 5 | \( 1 + 2.82T + 5T^{2} \) |
| 7 | \( 1 + 2.82T + 7T^{2} \) |
| 11 | \( 1 - 5.65T + 11T^{2} \) |
| 13 | \( 1 - 6T + 13T^{2} \) |
| 17 | \( 1 - 2.82T + 17T^{2} \) |
| 19 | \( 1 + 8.48T + 19T^{2} \) |
| 29 | \( 1 + 2T + 29T^{2} \) |
| 31 | \( 1 - 8T + 31T^{2} \) |
| 37 | \( 1 + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 - 2.82T + 43T^{2} \) |
| 47 | \( 1 + 8T + 47T^{2} \) |
| 53 | \( 1 + 8.48T + 53T^{2} \) |
| 59 | \( 1 + 4T + 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 - 8.48T + 67T^{2} \) |
| 71 | \( 1 + 8T + 71T^{2} \) |
| 73 | \( 1 - 6T + 73T^{2} \) |
| 79 | \( 1 - 2.82T + 79T^{2} \) |
| 83 | \( 1 + 5.65T + 83T^{2} \) |
| 89 | \( 1 + 2.82T + 89T^{2} \) |
| 97 | \( 1 - 5.65T + 97T^{2} \) |
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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
−7.64658135438301618809106208451, −6.65946541497931752144823284446, −6.38453136038632894148927603590, −5.93872193253636019596015585552, −4.57097329752742631798831532279, −4.07565733374831554159795470969, −3.60335718785474470521827675362, −3.05094368964784861351594620847, −1.71835201091011355915175368080, −0.69817369664604250788404374927,
0.69817369664604250788404374927, 1.71835201091011355915175368080, 3.05094368964784861351594620847, 3.60335718785474470521827675362, 4.07565733374831554159795470969, 4.57097329752742631798831532279, 5.93872193253636019596015585552, 6.38453136038632894148927603590, 6.65946541497931752144823284446, 7.64658135438301618809106208451
