L(s) = 1 | + 1.57·3-s + 4-s + 1.09·5-s + 1.49·9-s − 1.75·11-s + 1.57·12-s − 1.35·13-s + 1.72·15-s + 16-s − 1.97·19-s + 1.09·20-s + 0.196·25-s + 0.774·27-s − 2.77·33-s + 1.49·36-s − 2.13·39-s + 1.89·41-s − 1.75·44-s + 1.63·45-s + 1.57·48-s + 49-s − 1.35·52-s − 0.165·53-s − 1.92·55-s − 3.11·57-s − 1.35·59-s + 1.72·60-s + ⋯ |
L(s) = 1 | + 1.57·3-s + 4-s + 1.09·5-s + 1.49·9-s − 1.75·11-s + 1.57·12-s − 1.35·13-s + 1.72·15-s + 16-s − 1.97·19-s + 1.09·20-s + 0.196·25-s + 0.774·27-s − 2.77·33-s + 1.49·36-s − 2.13·39-s + 1.89·41-s − 1.75·44-s + 1.63·45-s + 1.57·48-s + 49-s − 1.35·52-s − 0.165·53-s − 1.92·55-s − 3.11·57-s − 1.35·59-s + 1.72·60-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
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.566632102\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.566632102\) |
\(L(1)\) |
|
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 | \( 1 - 1.57T + T^{2} \) |
| 5 | \( 1 - 1.09T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 + 1.75T + T^{2} \) |
| 13 | \( 1 + 1.35T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 1.97T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.89T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 + 0.165T + T^{2} \) |
| 59 | \( 1 + 1.35T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.89T + T^{2} \) |
| 71 | \( 1 + 0.803T + T^{2} \) |
| 73 | \( 1 - 0.490T + T^{2} \) |
| 79 | \( 1 - 1.89T + T^{2} \) |
| 83 | \( 1 + 1.97T + T^{2} \) |
| 89 | \( 1 + 0.803T + T^{2} \) |
| 97 | \( 1 + 0.165T + T^{2} \) |
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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
−9.218655803796362576661134299130, −8.259421474464773873954113139107, −7.75615995828977645189246879366, −7.08352622762939178666754053562, −6.13235518167138254066949013057, −5.30943384680449278747002826123, −4.22689874793899740606538129182, −2.88634755918027469564556805267, −2.39109456210827573813562317423, −2.00422563671398090515743335570,
2.00422563671398090515743335570, 2.39109456210827573813562317423, 2.88634755918027469564556805267, 4.22689874793899740606538129182, 5.30943384680449278747002826123, 6.13235518167138254066949013057, 7.08352622762939178666754053562, 7.75615995828977645189246879366, 8.259421474464773873954113139107, 9.218655803796362576661134299130