| L(s) = 1 | + 1.87·2-s − 1.53·3-s + 2.53·4-s − 2.87·6-s + 2.87·8-s + 1.34·9-s − 3.87·12-s − 0.347·13-s + 2.87·16-s + 2.53·18-s − 23-s − 4.41·24-s − 0.652·26-s − 0.532·27-s − 1.87·29-s − 1.87·31-s + 2.53·32-s + 3.41·36-s + 0.532·39-s + 1.53·41-s − 1.87·46-s − 0.347·47-s − 4.41·48-s + 49-s − 0.879·52-s − 54-s − 3.53·58-s + ⋯ |
| L(s) = 1 | + 1.87·2-s − 1.53·3-s + 2.53·4-s − 2.87·6-s + 2.87·8-s + 1.34·9-s − 3.87·12-s − 0.347·13-s + 2.87·16-s + 2.53·18-s − 23-s − 4.41·24-s − 0.652·26-s − 0.532·27-s − 1.87·29-s − 1.87·31-s + 2.53·32-s + 3.41·36-s + 0.532·39-s + 1.53·41-s − 1.87·46-s − 0.347·47-s − 4.41·48-s + 49-s − 0.879·52-s − 54-s − 3.53·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.662312689\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.662312689\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 2 | \( 1 - 1.87T + T^{2} \) |
| 3 | \( 1 + 1.53T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 11 | \( 1 - T^{2} \) |
| 13 | \( 1 + 0.347T + T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.87T + T^{2} \) |
| 31 | \( 1 + 1.87T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - 1.53T + T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 + 0.347T + T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - 0.347T + T^{2} \) |
| 73 | \( 1 - 1.87T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 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
−11.04508683171763821089151110398, −10.87963950797540412296852544066, −9.566974037216345423159262952153, −7.62290659815204742245646005111, −6.90099261831250610072036111005, −5.85312810210945507781523823885, −5.54940914813480352812913792073, −4.53258870454623338382412597617, −3.65652723499038769190938768506, −2.02466515105394899431729173924,
2.02466515105394899431729173924, 3.65652723499038769190938768506, 4.53258870454623338382412597617, 5.54940914813480352812913792073, 5.85312810210945507781523823885, 6.90099261831250610072036111005, 7.62290659815204742245646005111, 9.566974037216345423159262952153, 10.87963950797540412296852544066, 11.04508683171763821089151110398
