| L(s) = 1 | − 2.06·2-s + 2.27·4-s + 0.347·5-s − 4.33·7-s − 0.558·8-s − 0.718·10-s + 0.729·11-s − 5.40·13-s + 8.96·14-s − 3.38·16-s + 3.27·19-s + 0.789·20-s − 1.50·22-s − 3.55·23-s − 4.87·25-s + 11.1·26-s − 9.84·28-s + 4.38·29-s − 3.30·31-s + 8.11·32-s − 1.50·35-s − 2.87·37-s − 6.75·38-s − 0.194·40-s − 11.8·41-s + 3.74·43-s + 1.65·44-s + ⋯ |
| L(s) = 1 | − 1.46·2-s + 1.13·4-s + 0.155·5-s − 1.63·7-s − 0.197·8-s − 0.227·10-s + 0.220·11-s − 1.49·13-s + 2.39·14-s − 0.846·16-s + 0.750·19-s + 0.176·20-s − 0.321·22-s − 0.741·23-s − 0.975·25-s + 2.18·26-s − 1.86·28-s + 0.813·29-s − 0.593·31-s + 1.43·32-s − 0.254·35-s − 0.472·37-s − 1.09·38-s − 0.0306·40-s − 1.85·41-s + 0.571·43-s + 0.249·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2601 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.3551959301\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3551959301\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 17 | \( 1 \) |
| good | 2 | \( 1 + 2.06T + 2T^{2} \) |
| 5 | \( 1 - 0.347T + 5T^{2} \) |
| 7 | \( 1 + 4.33T + 7T^{2} \) |
| 11 | \( 1 - 0.729T + 11T^{2} \) |
| 13 | \( 1 + 5.40T + 13T^{2} \) |
| 19 | \( 1 - 3.27T + 19T^{2} \) |
| 23 | \( 1 + 3.55T + 23T^{2} \) |
| 29 | \( 1 - 4.38T + 29T^{2} \) |
| 31 | \( 1 + 3.30T + 31T^{2} \) |
| 37 | \( 1 + 2.87T + 37T^{2} \) |
| 41 | \( 1 + 11.8T + 41T^{2} \) |
| 43 | \( 1 - 3.74T + 43T^{2} \) |
| 47 | \( 1 + 0.476T + 47T^{2} \) |
| 53 | \( 1 - 10.1T + 53T^{2} \) |
| 59 | \( 1 + 5.29T + 59T^{2} \) |
| 61 | \( 1 - 3.54T + 61T^{2} \) |
| 67 | \( 1 + 1.55T + 67T^{2} \) |
| 71 | \( 1 - 8.96T + 71T^{2} \) |
| 73 | \( 1 + 4.09T + 73T^{2} \) |
| 79 | \( 1 - 0.812T + 79T^{2} \) |
| 83 | \( 1 + 7.21T + 83T^{2} \) |
| 89 | \( 1 - 9.77T + 89T^{2} \) |
| 97 | \( 1 + 1.77T + 97T^{2} \) |
| 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
−9.104541357510620443318767628896, −8.229692734629061248121743965667, −7.36071441113987177309172233652, −6.91592510037403480418945239184, −6.10351376711060952553768321756, −5.09365283102548073813281347185, −3.88055636929149595098882972955, −2.85903743046351064886108439926, −1.92897111751193044753882890818, −0.44537720398239109674726359902,
0.44537720398239109674726359902, 1.92897111751193044753882890818, 2.85903743046351064886108439926, 3.88055636929149595098882972955, 5.09365283102548073813281347185, 6.10351376711060952553768321756, 6.91592510037403480418945239184, 7.36071441113987177309172233652, 8.229692734629061248121743965667, 9.104541357510620443318767628896
