L(s) = 1 | + 1.53·2-s + 1.34·4-s + 1.53·5-s − 7-s + 0.532·8-s + 9-s + 2.34·10-s − 1.87·11-s − 1.53·14-s − 0.532·16-s + 1.53·18-s + 1.53·19-s + 2.06·20-s − 2.87·22-s − 23-s + 1.34·25-s − 1.34·28-s − 1.87·29-s − 1.34·32-s − 1.53·35-s + 1.34·36-s + 2·37-s + 2.34·38-s + 0.815·40-s − 1.87·41-s − 2.53·44-s + 1.53·45-s + ⋯ |
L(s) = 1 | + 1.53·2-s + 1.34·4-s + 1.53·5-s − 7-s + 0.532·8-s + 9-s + 2.34·10-s − 1.87·11-s − 1.53·14-s − 0.532·16-s + 1.53·18-s + 1.53·19-s + 2.06·20-s − 2.87·22-s − 23-s + 1.34·25-s − 1.34·28-s − 1.87·29-s − 1.34·32-s − 1.53·35-s + 1.34·36-s + 2·37-s + 2.34·38-s + 0.815·40-s − 1.87·41-s − 2.53·44-s + 1.53·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1399 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.638433010\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.638433010\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1399 | \( 1 - T \) |
good | 2 | \( 1 - 1.53T + T^{2} \) |
| 3 | \( 1 - T^{2} \) |
| 5 | \( 1 - 1.53T + T^{2} \) |
| 7 | \( 1 + T + T^{2} \) |
| 11 | \( 1 + 1.87T + T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - 1.53T + T^{2} \) |
| 23 | \( 1 + T + T^{2} \) |
| 29 | \( 1 + 1.87T + T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - 2T + T^{2} \) |
| 41 | \( 1 + 1.87T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 - 2T + T^{2} \) |
| 61 | \( 1 + T + T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 0.347T + T^{2} \) |
| 79 | \( 1 - 0.347T + T^{2} \) |
| 83 | \( 1 - 0.347T + T^{2} \) |
| 89 | \( 1 - 0.347T + T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−9.826199055948155339667691177078, −9.391792299018495697443129064086, −7.82137868188774127449798402128, −6.99121378451049749166723384384, −6.11017881766363102915930276149, −5.51916544941483688054507290524, −4.96438850065821880035329955178, −3.71561410187239841358929924722, −2.81253003133055267590828515228, −1.98344970866788644551150217601,
1.98344970866788644551150217601, 2.81253003133055267590828515228, 3.71561410187239841358929924722, 4.96438850065821880035329955178, 5.51916544941483688054507290524, 6.11017881766363102915930276149, 6.99121378451049749166723384384, 7.82137868188774127449798402128, 9.391792299018495697443129064086, 9.826199055948155339667691177078