L(s) = 1 | + 3-s + 5-s − 4.98·7-s + 9-s + 5.26·11-s + 6.98·13-s + 15-s + 0.280·17-s − 19-s − 4.98·21-s − 0.280·23-s + 25-s + 27-s − 3.26·29-s − 2.28·31-s + 5.26·33-s − 4.98·35-s − 2.98·37-s + 6.98·39-s + 0.738·41-s + 5.54·43-s + 45-s + 4.28·47-s + 17.8·49-s + 0.280·51-s + 5.71·53-s + 5.26·55-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.88·7-s + 0.333·9-s + 1.58·11-s + 1.93·13-s + 0.258·15-s + 0.0681·17-s − 0.229·19-s − 1.08·21-s − 0.0585·23-s + 0.200·25-s + 0.192·27-s − 0.605·29-s − 0.409·31-s + 0.915·33-s − 0.841·35-s − 0.489·37-s + 1.11·39-s + 0.115·41-s + 0.845·43-s + 0.149·45-s + 0.624·47-s + 2.54·49-s + 0.0393·51-s + 0.785·53-s + 0.709·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.562866072\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.562866072\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 - T \) |
| 19 | \( 1 + T \) |
good | 7 | \( 1 + 4.98T + 7T^{2} \) |
| 11 | \( 1 - 5.26T + 11T^{2} \) |
| 13 | \( 1 - 6.98T + 13T^{2} \) |
| 17 | \( 1 - 0.280T + 17T^{2} \) |
| 23 | \( 1 + 0.280T + 23T^{2} \) |
| 29 | \( 1 + 3.26T + 29T^{2} \) |
| 31 | \( 1 + 2.28T + 31T^{2} \) |
| 37 | \( 1 + 2.98T + 37T^{2} \) |
| 41 | \( 1 - 0.738T + 41T^{2} \) |
| 43 | \( 1 - 5.54T + 43T^{2} \) |
| 47 | \( 1 - 4.28T + 47T^{2} \) |
| 53 | \( 1 - 5.71T + 53T^{2} \) |
| 59 | \( 1 + 10.5T + 59T^{2} \) |
| 61 | \( 1 + 2.24T + 61T^{2} \) |
| 67 | \( 1 + 67T^{2} \) |
| 71 | \( 1 + 14.5T + 71T^{2} \) |
| 73 | \( 1 - 5.43T + 73T^{2} \) |
| 79 | \( 1 - 16.4T + 79T^{2} \) |
| 83 | \( 1 + 10.8T + 83T^{2} \) |
| 89 | \( 1 - 16.6T + 89T^{2} \) |
| 97 | \( 1 - 2.98T + 97T^{2} \) |
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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
−8.690499728016841791324663769881, −7.49578735522363257772702704975, −6.70936327852825930391181139888, −6.20671825135323872467225093757, −5.77788073950248261964363644484, −4.22043161759661705658989496702, −3.63400611632924262343914540835, −3.15590086343900222633230438112, −1.92877499568284145264679222143, −0.901414376203660423871361901039,
0.901414376203660423871361901039, 1.92877499568284145264679222143, 3.15590086343900222633230438112, 3.63400611632924262343914540835, 4.22043161759661705658989496702, 5.77788073950248261964363644484, 6.20671825135323872467225093757, 6.70936327852825930391181139888, 7.49578735522363257772702704975, 8.690499728016841791324663769881