| L(s) = 1 | − 1.17·2-s − 1.90·3-s + 0.381·4-s − 5-s + 2.23·6-s + 0.726·8-s + 2.61·9-s + 1.17·10-s + 11-s − 0.726·12-s + 1.90·15-s − 1.23·16-s − 3.07·18-s − 0.618·19-s − 0.381·20-s − 1.17·22-s − 1.38·24-s + 25-s − 3.07·27-s − 1.61·29-s − 2.23·30-s + 0.618·31-s + 0.726·32-s − 1.90·33-s + 0.999·36-s + 0.726·38-s − 0.726·40-s + ⋯ |
| L(s) = 1 | − 1.17·2-s − 1.90·3-s + 0.381·4-s − 5-s + 2.23·6-s + 0.726·8-s + 2.61·9-s + 1.17·10-s + 11-s − 0.726·12-s + 1.90·15-s − 1.23·16-s − 3.07·18-s − 0.618·19-s − 0.381·20-s − 1.17·22-s − 1.38·24-s + 25-s − 3.07·27-s − 1.61·29-s − 2.23·30-s + 0.618·31-s + 0.726·32-s − 1.90·33-s + 0.999·36-s + 0.726·38-s − 0.726·40-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2255 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2255 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2197472886\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2197472886\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| good | 2 | \( 1 + 1.17T + T^{2} \) |
| 3 | \( 1 + 1.90T + T^{2} \) |
| 7 | \( 1 - T^{2} \) |
| 13 | \( 1 - T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + 0.618T + T^{2} \) |
| 23 | \( 1 - T^{2} \) |
| 29 | \( 1 + 1.61T + T^{2} \) |
| 31 | \( 1 - 0.618T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 + T^{2} \) |
| 53 | \( 1 - 1.17T + T^{2} \) |
| 59 | \( 1 + 1.61T + T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 1.17T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 - 1.90T + T^{2} \) |
| 79 | \( 1 + 1.61T + T^{2} \) |
| 83 | \( 1 + 1.90T + T^{2} \) |
| 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
−9.280747484079235434770616997124, −8.503637862890635975267085129835, −7.53088295085695331499580276312, −7.00862418553826629237081144915, −6.31697034886179590202706834927, −5.30784905773811018908255880403, −4.42745336522265578914158503030, −3.86008178723614231462539438165, −1.68149246780678072125328813077, −0.61599312955875826072061789363,
0.61599312955875826072061789363, 1.68149246780678072125328813077, 3.86008178723614231462539438165, 4.42745336522265578914158503030, 5.30784905773811018908255880403, 6.31697034886179590202706834927, 7.00862418553826629237081144915, 7.53088295085695331499580276312, 8.503637862890635975267085129835, 9.280747484079235434770616997124
