L(s) = 1 | + 2.56·2-s + 1.79·3-s + 4.56·4-s + 4.59·6-s + 3.80·7-s + 6.57·8-s + 0.213·9-s + 8.18·12-s − 4.49·13-s + 9.73·14-s + 7.71·16-s + 5.01·17-s + 0.546·18-s − 3.46·19-s + 6.81·21-s + 0.105·23-s + 11.7·24-s − 11.5·26-s − 4.99·27-s + 17.3·28-s − 2.35·29-s + 4.74·31-s + 6.62·32-s + 12.8·34-s + 0.974·36-s − 5.60·37-s − 8.86·38-s + ⋯ |
L(s) = 1 | + 1.81·2-s + 1.03·3-s + 2.28·4-s + 1.87·6-s + 1.43·7-s + 2.32·8-s + 0.0711·9-s + 2.36·12-s − 1.24·13-s + 2.60·14-s + 1.92·16-s + 1.21·17-s + 0.128·18-s − 0.793·19-s + 1.48·21-s + 0.0220·23-s + 2.40·24-s − 2.26·26-s − 0.961·27-s + 3.27·28-s − 0.436·29-s + 0.852·31-s + 1.17·32-s + 2.20·34-s + 0.162·36-s − 0.920·37-s − 1.43·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3025 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(8.863714600\) |
\(L(\frac12)\) |
\(\approx\) |
\(8.863714600\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 - 2.56T + 2T^{2} \) |
| 3 | \( 1 - 1.79T + 3T^{2} \) |
| 7 | \( 1 - 3.80T + 7T^{2} \) |
| 13 | \( 1 + 4.49T + 13T^{2} \) |
| 17 | \( 1 - 5.01T + 17T^{2} \) |
| 19 | \( 1 + 3.46T + 19T^{2} \) |
| 23 | \( 1 - 0.105T + 23T^{2} \) |
| 29 | \( 1 + 2.35T + 29T^{2} \) |
| 31 | \( 1 - 4.74T + 31T^{2} \) |
| 37 | \( 1 + 5.60T + 37T^{2} \) |
| 41 | \( 1 - 0.916T + 41T^{2} \) |
| 43 | \( 1 + 4.46T + 43T^{2} \) |
| 47 | \( 1 - 1.25T + 47T^{2} \) |
| 53 | \( 1 + 5.49T + 53T^{2} \) |
| 59 | \( 1 + 1.18T + 59T^{2} \) |
| 61 | \( 1 + 7.32T + 61T^{2} \) |
| 67 | \( 1 - 7.84T + 67T^{2} \) |
| 71 | \( 1 + 2.18T + 71T^{2} \) |
| 73 | \( 1 - 8.95T + 73T^{2} \) |
| 79 | \( 1 - 14.3T + 79T^{2} \) |
| 83 | \( 1 + 4.71T + 83T^{2} \) |
| 89 | \( 1 + 0.172T + 89T^{2} \) |
| 97 | \( 1 + 4.46T + 97T^{2} \) |
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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
−8.383064726851346365536932811799, −7.83136376566108758766397130219, −7.24955699921671542523434079250, −6.24563701523679692079262755761, −5.27346456027390480705618358518, −4.88146051534187280322077360947, −4.01491377667170165795874658276, −3.20118767049142718885447908731, −2.39320774559316375551510409920, −1.71120679492796465412732604213,
1.71120679492796465412732604213, 2.39320774559316375551510409920, 3.20118767049142718885447908731, 4.01491377667170165795874658276, 4.88146051534187280322077360947, 5.27346456027390480705618358518, 6.24563701523679692079262755761, 7.24955699921671542523434079250, 7.83136376566108758766397130219, 8.383064726851346365536932811799