| L(s) = 1 | + 1.89·3-s + 4-s − 1.75·5-s + 2.57·9-s + 0.490·11-s + 1.89·12-s − 0.803·13-s − 3.32·15-s + 16-s − 0.165·19-s − 1.75·20-s + 2.09·25-s + 2.98·27-s + 0.928·33-s + 2.57·36-s − 1.51·39-s − 1.97·41-s + 0.490·44-s − 4.53·45-s + 1.89·48-s + 49-s − 0.803·52-s − 1.35·53-s − 0.863·55-s − 0.312·57-s − 0.803·59-s − 3.32·60-s + ⋯ |
| L(s) = 1 | + 1.89·3-s + 4-s − 1.75·5-s + 2.57·9-s + 0.490·11-s + 1.89·12-s − 0.803·13-s − 3.32·15-s + 16-s − 0.165·19-s − 1.75·20-s + 2.09·25-s + 2.98·27-s + 0.928·33-s + 2.57·36-s − 1.51·39-s − 1.97·41-s + 0.490·44-s − 4.53·45-s + 1.89·48-s + 49-s − 0.803·52-s − 1.35·53-s − 0.863·55-s − 0.312·57-s − 0.803·59-s − 3.32·60-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2339 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(2.188518548\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.188518548\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2339 | \( 1+O(T) \) |
| good | 2 | \( 1 - T^{2} \) |
| 3 | \( 1 - 1.89T + T^{2} \) |
| 5 | \( 1 + 1.75T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - 0.490T + T^{2} \) |
| 13 | \( 1 + 0.803T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 0.165T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 + 1.97T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 1.35T + T^{2} \) |
| 59 | \( 1 + 0.803T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 + 1.97T + T^{2} \) |
| 71 | \( 1 - 1.09T + T^{2} \) |
| 73 | \( 1 - 1.57T + T^{2} \) |
| 79 | \( 1 + 1.97T + T^{2} \) |
| 83 | \( 1 + 0.165T + T^{2} \) |
| 89 | \( 1 - 1.09T + T^{2} \) |
| 97 | \( 1 + 1.35T + T^{2} \) |
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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
−8.903741111214100567142020735885, −8.272362190191611486478294557762, −7.68367598085877528634306269807, −7.22129322971635701715303351381, −6.57297058869392401919195625842, −4.83831822167550826101787138264, −3.97502395873176381531425337539, −3.33890241536102076590500829042, −2.67659744825457391585673361345, −1.56883179345890388832480677486,
1.56883179345890388832480677486, 2.67659744825457391585673361345, 3.33890241536102076590500829042, 3.97502395873176381531425337539, 4.83831822167550826101787138264, 6.57297058869392401919195625842, 7.22129322971635701715303351381, 7.68367598085877528634306269807, 8.272362190191611486478294557762, 8.903741111214100567142020735885
