L(s) = 1 | + 0.556·2-s + 2.94·3-s − 1.69·4-s + 1.63·6-s − 7-s − 2.05·8-s + 5.65·9-s − 5.83·11-s − 4.97·12-s + 4.53·13-s − 0.556·14-s + 2.24·16-s + 3.00·17-s + 3.14·18-s + 3.78·19-s − 2.94·21-s − 3.24·22-s − 23-s − 6.03·24-s + 2.52·26-s + 7.80·27-s + 1.69·28-s − 6.57·29-s + 6.03·31-s + 5.35·32-s − 17.1·33-s + 1.66·34-s + ⋯ |
L(s) = 1 | + 0.393·2-s + 1.69·3-s − 0.845·4-s + 0.667·6-s − 0.377·7-s − 0.725·8-s + 1.88·9-s − 1.75·11-s − 1.43·12-s + 1.25·13-s − 0.148·14-s + 0.560·16-s + 0.727·17-s + 0.741·18-s + 0.867·19-s − 0.641·21-s − 0.691·22-s − 0.208·23-s − 1.23·24-s + 0.494·26-s + 1.50·27-s + 0.319·28-s − 1.22·29-s + 1.08·31-s + 0.945·32-s − 2.98·33-s + 0.286·34-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
\(L(1)\) |
\(\approx\) |
\(3.339782062\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.339782062\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 - 0.556T + 2T^{2} \) |
| 3 | \( 1 - 2.94T + 3T^{2} \) |
| 11 | \( 1 + 5.83T + 11T^{2} \) |
| 13 | \( 1 - 4.53T + 13T^{2} \) |
| 17 | \( 1 - 3.00T + 17T^{2} \) |
| 19 | \( 1 - 3.78T + 19T^{2} \) |
| 29 | \( 1 + 6.57T + 29T^{2} \) |
| 31 | \( 1 - 6.03T + 31T^{2} \) |
| 37 | \( 1 - 2.03T + 37T^{2} \) |
| 41 | \( 1 - 8.22T + 41T^{2} \) |
| 43 | \( 1 - 8.07T + 43T^{2} \) |
| 47 | \( 1 - 8.34T + 47T^{2} \) |
| 53 | \( 1 - 5.57T + 53T^{2} \) |
| 59 | \( 1 + 11.3T + 59T^{2} \) |
| 61 | \( 1 + 1.32T + 61T^{2} \) |
| 67 | \( 1 + 0.934T + 67T^{2} \) |
| 71 | \( 1 + 0.463T + 71T^{2} \) |
| 73 | \( 1 - 13.8T + 73T^{2} \) |
| 79 | \( 1 - 5.34T + 79T^{2} \) |
| 83 | \( 1 - 9.37T + 83T^{2} \) |
| 89 | \( 1 + 15.5T + 89T^{2} \) |
| 97 | \( 1 - 3.55T + 97T^{2} \) |
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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
−8.368419072349861593715026207237, −7.86552930907815291687929500099, −7.40584784638680826446252242318, −6.00902984088498337262372443112, −5.44551531748727249918709582504, −4.40240692935951100666326187540, −3.68534627748355986454365424458, −3.07262031835963364585247190515, −2.39890561095783072446320816373, −0.933831532304327378021629065603,
0.933831532304327378021629065603, 2.39890561095783072446320816373, 3.07262031835963364585247190515, 3.68534627748355986454365424458, 4.40240692935951100666326187540, 5.44551531748727249918709582504, 6.00902984088498337262372443112, 7.40584784638680826446252242318, 7.86552930907815291687929500099, 8.368419072349861593715026207237