L(s) = 1 | + 2.76·3-s − 2.37·5-s + 2.36·7-s + 4.64·9-s − 1.30·11-s − 3.45·13-s − 6.57·15-s − 3.40·17-s − 3.93·19-s + 6.53·21-s − 0.710·23-s + 0.657·25-s + 4.56·27-s + 5.88·29-s − 3.60·31-s − 3.61·33-s − 5.62·35-s + 7.19·37-s − 9.55·39-s + 7.35·41-s − 3.43·43-s − 11.0·45-s + 0.666·47-s − 1.41·49-s − 9.41·51-s + 1.77·53-s + 3.11·55-s + ⋯ |
L(s) = 1 | + 1.59·3-s − 1.06·5-s + 0.893·7-s + 1.54·9-s − 0.394·11-s − 0.958·13-s − 1.69·15-s − 0.825·17-s − 0.901·19-s + 1.42·21-s − 0.148·23-s + 0.131·25-s + 0.878·27-s + 1.09·29-s − 0.647·31-s − 0.629·33-s − 0.950·35-s + 1.18·37-s − 1.53·39-s + 1.14·41-s − 0.523·43-s − 1.64·45-s + 0.0972·47-s − 0.202·49-s − 1.31·51-s + 0.243·53-s + 0.419·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8048 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 503 | \( 1 - T \) |
good | 3 | \( 1 - 2.76T + 3T^{2} \) |
| 5 | \( 1 + 2.37T + 5T^{2} \) |
| 7 | \( 1 - 2.36T + 7T^{2} \) |
| 11 | \( 1 + 1.30T + 11T^{2} \) |
| 13 | \( 1 + 3.45T + 13T^{2} \) |
| 17 | \( 1 + 3.40T + 17T^{2} \) |
| 19 | \( 1 + 3.93T + 19T^{2} \) |
| 23 | \( 1 + 0.710T + 23T^{2} \) |
| 29 | \( 1 - 5.88T + 29T^{2} \) |
| 31 | \( 1 + 3.60T + 31T^{2} \) |
| 37 | \( 1 - 7.19T + 37T^{2} \) |
| 41 | \( 1 - 7.35T + 41T^{2} \) |
| 43 | \( 1 + 3.43T + 43T^{2} \) |
| 47 | \( 1 - 0.666T + 47T^{2} \) |
| 53 | \( 1 - 1.77T + 53T^{2} \) |
| 59 | \( 1 + 4.65T + 59T^{2} \) |
| 61 | \( 1 - 1.08T + 61T^{2} \) |
| 67 | \( 1 + 4.94T + 67T^{2} \) |
| 71 | \( 1 + 9.87T + 71T^{2} \) |
| 73 | \( 1 + 5.89T + 73T^{2} \) |
| 79 | \( 1 + 5.05T + 79T^{2} \) |
| 83 | \( 1 - 0.501T + 83T^{2} \) |
| 89 | \( 1 + 14.0T + 89T^{2} \) |
| 97 | \( 1 + 0.210T + 97T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.63905984058363412063756589230, −7.24579781696682213802995637189, −6.27521780894991739560589435883, −5.08548232854992960596935143013, −4.27746343895723798451517604638, −4.09854970808164510097708431493, −2.89641350083970004794848066747, −2.46843210895920392019481007562, −1.54705744694947460437669292511, 0,
1.54705744694947460437669292511, 2.46843210895920392019481007562, 2.89641350083970004794848066747, 4.09854970808164510097708431493, 4.27746343895723798451517604638, 5.08548232854992960596935143013, 6.27521780894991739560589435883, 7.24579781696682213802995637189, 7.63905984058363412063756589230