L(s) = 1 | − 2-s + 4-s − 0.685·5-s − 0.816·7-s − 8-s + 0.685·10-s − 11-s − 4.21·13-s + 0.816·14-s + 16-s + 4.97·17-s + 19-s − 0.685·20-s + 22-s − 6.21·23-s − 4.53·25-s + 4.21·26-s − 0.816·28-s − 6.90·29-s + 9.10·31-s − 32-s − 4.97·34-s + 0.559·35-s + 10.9·37-s − 38-s + 0.685·40-s − 3.70·41-s + ⋯ |
L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.306·5-s − 0.308·7-s − 0.353·8-s + 0.216·10-s − 0.301·11-s − 1.16·13-s + 0.218·14-s + 0.250·16-s + 1.20·17-s + 0.229·19-s − 0.153·20-s + 0.213·22-s − 1.29·23-s − 0.906·25-s + 0.826·26-s − 0.154·28-s − 1.28·29-s + 1.63·31-s − 0.176·32-s − 0.852·34-s + 0.0945·35-s + 1.79·37-s − 0.162·38-s + 0.108·40-s − 0.578·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3762 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.8669713461\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8669713461\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + T \) |
| 19 | \( 1 - T \) |
good | 5 | \( 1 + 0.685T + 5T^{2} \) |
| 7 | \( 1 + 0.816T + 7T^{2} \) |
| 13 | \( 1 + 4.21T + 13T^{2} \) |
| 17 | \( 1 - 4.97T + 17T^{2} \) |
| 23 | \( 1 + 6.21T + 23T^{2} \) |
| 29 | \( 1 + 6.90T + 29T^{2} \) |
| 31 | \( 1 - 9.10T + 31T^{2} \) |
| 37 | \( 1 - 10.9T + 37T^{2} \) |
| 41 | \( 1 + 3.70T + 41T^{2} \) |
| 43 | \( 1 - 7.53T + 43T^{2} \) |
| 47 | \( 1 + 6.21T + 47T^{2} \) |
| 53 | \( 1 + 0.885T + 53T^{2} \) |
| 59 | \( 1 + 1.93T + 59T^{2} \) |
| 61 | \( 1 - 14.0T + 61T^{2} \) |
| 67 | \( 1 + 4.34T + 67T^{2} \) |
| 71 | \( 1 + 2.44T + 71T^{2} \) |
| 73 | \( 1 - 9.96T + 73T^{2} \) |
| 79 | \( 1 + 5.21T + 79T^{2} \) |
| 83 | \( 1 + 3.70T + 83T^{2} \) |
| 89 | \( 1 - 4.41T + 89T^{2} \) |
| 97 | \( 1 + 7.12T + 97T^{2} \) |
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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
−8.311872278348695484493576370024, −7.77797896878271647294884485273, −7.37719992280559496209814452596, −6.30431347700291214045282534364, −5.70597807455591905904777393631, −4.72931564101938128586056462778, −3.75789667472945037327957718751, −2.84049371640680141081287496365, −1.95468438182436647504789334509, −0.58198591478198864242726901677,
0.58198591478198864242726901677, 1.95468438182436647504789334509, 2.84049371640680141081287496365, 3.75789667472945037327957718751, 4.72931564101938128586056462778, 5.70597807455591905904777393631, 6.30431347700291214045282534364, 7.37719992280559496209814452596, 7.77797896878271647294884485273, 8.311872278348695484493576370024