L(s) = 1 | − 1.91·2-s + 1.65·3-s + 1.65·4-s − 3.16·6-s + 0.656·8-s − 0.255·9-s + 4.48·11-s + 2.74·12-s + 6.48·13-s − 4.56·16-s + 1.08·17-s + 0.488·18-s − 2.22·19-s − 8.56·22-s + 1.25·23-s + 1.08·24-s − 12.3·26-s − 5.39·27-s − 3.59·29-s − 2.59·31-s + 7.42·32-s + 7.42·33-s − 2.08·34-s − 0.423·36-s + 7.16·37-s + 4.25·38-s + 10.7·39-s + ⋯ |
L(s) = 1 | − 1.35·2-s + 0.956·3-s + 0.828·4-s − 1.29·6-s + 0.232·8-s − 0.0852·9-s + 1.35·11-s + 0.792·12-s + 1.79·13-s − 1.14·16-s + 0.263·17-s + 0.115·18-s − 0.510·19-s − 1.82·22-s + 0.261·23-s + 0.222·24-s − 2.43·26-s − 1.03·27-s − 0.668·29-s − 0.466·31-s + 1.31·32-s + 1.29·33-s − 0.356·34-s − 0.0705·36-s + 1.17·37-s + 0.690·38-s + 1.71·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.261105209\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.261105209\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + 1.91T + 2T^{2} \) |
| 3 | \( 1 - 1.65T + 3T^{2} \) |
| 11 | \( 1 - 4.48T + 11T^{2} \) |
| 13 | \( 1 - 6.48T + 13T^{2} \) |
| 17 | \( 1 - 1.08T + 17T^{2} \) |
| 19 | \( 1 + 2.22T + 19T^{2} \) |
| 23 | \( 1 - 1.25T + 23T^{2} \) |
| 29 | \( 1 + 3.59T + 29T^{2} \) |
| 31 | \( 1 + 2.59T + 31T^{2} \) |
| 37 | \( 1 - 7.16T + 37T^{2} \) |
| 41 | \( 1 + 3.88T + 41T^{2} \) |
| 43 | \( 1 - 3.08T + 43T^{2} \) |
| 47 | \( 1 - 6.43T + 47T^{2} \) |
| 53 | \( 1 + 3.05T + 53T^{2} \) |
| 59 | \( 1 - 11.3T + 59T^{2} \) |
| 61 | \( 1 + 2.59T + 61T^{2} \) |
| 67 | \( 1 - 5.31T + 67T^{2} \) |
| 71 | \( 1 - 4.19T + 71T^{2} \) |
| 73 | \( 1 + 6.62T + 73T^{2} \) |
| 79 | \( 1 + 14.2T + 79T^{2} \) |
| 83 | \( 1 - 9.59T + 83T^{2} \) |
| 89 | \( 1 - 12.0T + 89T^{2} \) |
| 97 | \( 1 - 4.67T + 97T^{2} \) |
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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
−9.291038613946797048489790364984, −8.962075992499942653223319947683, −8.383341620945627530388290523659, −7.64675696514773221316472872669, −6.68446372249811372107098263099, −5.83866747426827904559376333103, −4.18830331520501419060002631809, −3.46169801972633001087214736367, −2.06712828106971264844436686403, −1.05583869135541896334512515144,
1.05583869135541896334512515144, 2.06712828106971264844436686403, 3.46169801972633001087214736367, 4.18830331520501419060002631809, 5.83866747426827904559376333103, 6.68446372249811372107098263099, 7.64675696514773221316472872669, 8.383341620945627530388290523659, 8.962075992499942653223319947683, 9.291038613946797048489790364984