After the Riemann zeta function, the analytic conductor of this L-function is the smallest among L-functions of degree 1.
L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s + 11-s − 12-s − 13-s + 14-s + 16-s − 17-s − 18-s + 19-s + 21-s − 22-s − 23-s + 24-s + 26-s − 27-s − 28-s + 29-s + 31-s − 32-s − 33-s + 34-s + ⋯ |
L(s) = 1 | − 2-s − 3-s + 4-s + 6-s − 7-s − 8-s + 9-s + 11-s − 12-s − 13-s + 14-s + 16-s − 17-s − 18-s + 19-s + 21-s − 22-s − 23-s + 24-s + 26-s − 27-s − 28-s + 29-s + 31-s − 32-s − 33-s + 34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.2317509475\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.2317509475\) |
\(L(1)\) |
\(\approx\) |
\(0.4304089409\) |
\(L(1)\) |
\(\approx\) |
\(0.4304089409\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 - T \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - T \) |
| 89 | \( 1 + T \) |
| 97 | \( 1 - T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−55.58928033540481015096458899849, −53.83044519544216335105426902233, −52.1259022313169741886041306959, −51.08775192674649135525825720413, −48.34566182106784617654820130449, −46.49272715949140534533919935167, −45.4273000827822893888415619096, −44.03129006144169504470090805842, −41.84243854579169430850930688531, −39.56057294640318170505509505995, −38.12918472143653185015141827037, −35.86863837181227459459504863887, −34.728812978904808674143729833981, −33.00045600687051436794975917721, −29.70790935048096556923098651865, −28.46103510017752247518697827232, −26.77609594800414011652357496527, −24.58846621740819520765626997608, −22.227405454459410911877624963081, −19.54073262278475025037869002299, −17.566994292325555202701595268144, −16.03382112838423567459325378224, −11.95884562608351453026565868826, −9.831444432886669616348321347458, −6.64845334472771471612327845997,
6.64845334472771471612327845997, 9.831444432886669616348321347458, 11.95884562608351453026565868826, 16.03382112838423567459325378224, 17.566994292325555202701595268144, 19.54073262278475025037869002299, 22.227405454459410911877624963081, 24.58846621740819520765626997608, 26.77609594800414011652357496527, 28.46103510017752247518697827232, 29.70790935048096556923098651865, 33.00045600687051436794975917721, 34.728812978904808674143729833981, 35.86863837181227459459504863887, 38.12918472143653185015141827037, 39.56057294640318170505509505995, 41.84243854579169430850930688531, 44.03129006144169504470090805842, 45.4273000827822893888415619096, 46.49272715949140534533919935167, 48.34566182106784617654820130449, 51.08775192674649135525825720413, 52.1259022313169741886041306959, 53.83044519544216335105426902233, 55.58928033540481015096458899849