L(s) = 1 | + 2.60·2-s − 2.93·3-s + 4.80·4-s − 7.64·6-s − 7-s + 7.30·8-s + 5.58·9-s + 0.264·11-s − 14.0·12-s − 1.35·13-s − 2.60·14-s + 9.45·16-s − 5.30·17-s + 14.5·18-s − 4.34·19-s + 2.93·21-s + 0.690·22-s − 23-s − 21.4·24-s − 3.53·26-s − 7.57·27-s − 4.80·28-s − 5.48·29-s + 8.13·31-s + 10.0·32-s − 0.776·33-s − 13.8·34-s + ⋯ |
L(s) = 1 | + 1.84·2-s − 1.69·3-s + 2.40·4-s − 3.11·6-s − 0.377·7-s + 2.58·8-s + 1.86·9-s + 0.0798·11-s − 4.06·12-s − 0.376·13-s − 0.697·14-s + 2.36·16-s − 1.28·17-s + 3.43·18-s − 0.995·19-s + 0.639·21-s + 0.147·22-s − 0.208·23-s − 4.37·24-s − 0.693·26-s − 1.45·27-s − 0.907·28-s − 1.01·29-s + 1.46·31-s + 1.77·32-s − 0.135·33-s − 2.37·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 23 | \( 1 + T \) |
good | 2 | \( 1 - 2.60T + 2T^{2} \) |
| 3 | \( 1 + 2.93T + 3T^{2} \) |
| 11 | \( 1 - 0.264T + 11T^{2} \) |
| 13 | \( 1 + 1.35T + 13T^{2} \) |
| 17 | \( 1 + 5.30T + 17T^{2} \) |
| 19 | \( 1 + 4.34T + 19T^{2} \) |
| 29 | \( 1 + 5.48T + 29T^{2} \) |
| 31 | \( 1 - 8.13T + 31T^{2} \) |
| 37 | \( 1 + 5.32T + 37T^{2} \) |
| 41 | \( 1 - 2.17T + 41T^{2} \) |
| 43 | \( 1 - 2.38T + 43T^{2} \) |
| 47 | \( 1 + 11.7T + 47T^{2} \) |
| 53 | \( 1 - 8.02T + 53T^{2} \) |
| 59 | \( 1 + 9.46T + 59T^{2} \) |
| 61 | \( 1 - 3.95T + 61T^{2} \) |
| 67 | \( 1 + 9.20T + 67T^{2} \) |
| 71 | \( 1 + 9.36T + 71T^{2} \) |
| 73 | \( 1 + 6.26T + 73T^{2} \) |
| 79 | \( 1 + 7.93T + 79T^{2} \) |
| 83 | \( 1 - 2.13T + 83T^{2} \) |
| 89 | \( 1 - 14.4T + 89T^{2} \) |
| 97 | \( 1 + 15.4T + 97T^{2} \) |
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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
−7.46674585537375713627576665998, −6.83774967518416063436072043295, −6.25085816115242032731759939520, −5.89743866184251389765808339773, −4.95026161911545717071476711284, −4.54138124957736009357291580123, −3.84491415837763389432257558947, −2.67471787605060699143680643281, −1.67698804322615112842272573469, 0,
1.67698804322615112842272573469, 2.67471787605060699143680643281, 3.84491415837763389432257558947, 4.54138124957736009357291580123, 4.95026161911545717071476711284, 5.89743866184251389765808339773, 6.25085816115242032731759939520, 6.83774967518416063436072043295, 7.46674585537375713627576665998