L(s) = 1 | − 0.439·2-s + 3.43·3-s − 1.80·4-s − 2.66·5-s − 1.50·6-s + 1.67·8-s + 8.78·9-s + 1.17·10-s + 4.14·11-s − 6.20·12-s − 2.55·13-s − 9.15·15-s + 2.88·16-s − 1.58·17-s − 3.85·18-s + 1.76·19-s + 4.82·20-s − 1.81·22-s − 2.34·23-s + 5.73·24-s + 2.11·25-s + 1.12·26-s + 19.8·27-s + 5.28·29-s + 4.01·30-s + 31-s − 4.60·32-s + ⋯ |
L(s) = 1 | − 0.310·2-s + 1.98·3-s − 0.903·4-s − 1.19·5-s − 0.615·6-s + 0.591·8-s + 2.92·9-s + 0.370·10-s + 1.24·11-s − 1.79·12-s − 0.708·13-s − 2.36·15-s + 0.720·16-s − 0.383·17-s − 0.908·18-s + 0.405·19-s + 1.07·20-s − 0.387·22-s − 0.489·23-s + 1.17·24-s + 0.422·25-s + 0.219·26-s + 3.81·27-s + 0.981·29-s + 0.733·30-s + 0.179·31-s − 0.814·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1519 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.141715184\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.141715184\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 31 | \( 1 - T \) |
good | 2 | \( 1 + 0.439T + 2T^{2} \) |
| 3 | \( 1 - 3.43T + 3T^{2} \) |
| 5 | \( 1 + 2.66T + 5T^{2} \) |
| 11 | \( 1 - 4.14T + 11T^{2} \) |
| 13 | \( 1 + 2.55T + 13T^{2} \) |
| 17 | \( 1 + 1.58T + 17T^{2} \) |
| 19 | \( 1 - 1.76T + 19T^{2} \) |
| 23 | \( 1 + 2.34T + 23T^{2} \) |
| 29 | \( 1 - 5.28T + 29T^{2} \) |
| 37 | \( 1 - 3.67T + 37T^{2} \) |
| 41 | \( 1 + 0.0853T + 41T^{2} \) |
| 43 | \( 1 - 8.97T + 43T^{2} \) |
| 47 | \( 1 - 1.04T + 47T^{2} \) |
| 53 | \( 1 - 0.624T + 53T^{2} \) |
| 59 | \( 1 - 5.68T + 59T^{2} \) |
| 61 | \( 1 - 0.870T + 61T^{2} \) |
| 67 | \( 1 + 6.37T + 67T^{2} \) |
| 71 | \( 1 - 8.24T + 71T^{2} \) |
| 73 | \( 1 - 3.25T + 73T^{2} \) |
| 79 | \( 1 - 15.9T + 79T^{2} \) |
| 83 | \( 1 - 4.72T + 83T^{2} \) |
| 89 | \( 1 - 12.9T + 89T^{2} \) |
| 97 | \( 1 + 4.68T + 97T^{2} \) |
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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
−9.326718882601765316719046071335, −8.657591226070609921724740005217, −7.995671901813995112824751201224, −7.54616943622246922657724413518, −6.66684400221528423922567496346, −4.78339198851491493439463353257, −4.05713044477312009858089276995, −3.65323204042147179559372418428, −2.48165749576613668662979791415, −1.07335825998928812379807327641,
1.07335825998928812379807327641, 2.48165749576613668662979791415, 3.65323204042147179559372418428, 4.05713044477312009858089276995, 4.78339198851491493439463353257, 6.66684400221528423922567496346, 7.54616943622246922657724413518, 7.995671901813995112824751201224, 8.657591226070609921724740005217, 9.326718882601765316719046071335